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Signatures in algebra, topology and dynamics

Citation for published version: Ghys, E & Ranicki, A 2016, 'Signatures in algebra, topology and dynamics' Ensaios Matemáticos, vol. 30, pp. 1-173.

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Download date: 05. Apr. 2019 Signatures in algebra, topology and dynamics

Etienne´ Ghys, Andrew Ranicki 31st December, 2015 arXiv:1512.09258v1 [math.AT] 31 Dec 2015 E.G A.R. UMPA UMR 5669 CNRS School of Mathematics, ENS Lyon University of Edinburgh Site Monod James Clerk Maxwell Building 46 All´eed’Italie Peter Guthrie Tait Road 69364 Lyon Edinburgh EH9 3FD France Scotland, UK [email protected] [email protected]

1 Contents

1 Algebra 6 1.1 The basic definitions ...... 6 1.2 Linear congruence ...... 7 1.3 The signature of a symmetric matrix ...... 11 1.4 Orthogonal congruence ...... 13 1.5 Continued fractions, tridiagonal matrices and signatures . . . 22 1.6 Sturm’s theorem and its reformulation by Sylvester ...... 27 1.7 Witt groups of fields ...... 36 1.8 The Witt groups of the function field R(X), ordered fields and Q...... 42 2 Topology 53 2.1 Even-dimensional manifolds ...... 53 2.2 Cobordism ...... 58 2.3 Odd-dimensional manifolds ...... 61 2.4 The union of manifolds with boundary; Novikov additivity and Wall nonadditivity of the signature ...... 65 2.5 Plumbing: from quadratic forms to manifolds ...... 72

3 Knots, links, braids and signatures 78 3.1 Knots, links, Seifert surfaces and complements ...... 78 3.2 Signatures of knots : a tale from the sixties ...... 83 3.3 Two great papers by Milnor and the modern approach . . . . 84 3.4 Braids ...... 89 3.5 Knots and signatures in higher dimensions ...... 93

4 The symplectic group and the Maslov class 95 4.1 A very brief history of the symplectic group ...... 95 4.2 The Maslov class ...... 98 4.3 A short recollection of group cohomology ...... 100 4.4 The topology of Λn ...... 101 4.5 Maslov indices and the universal cover of Λn ...... 102 4.6 The Maslov class as a bounded cohomology class ...... 105 4.7 Picture in the case n =2...... 109

2 5 Dynamics 111 5.1 The Calabi and Ruelle invariant ...... 111 5.2 The Conley-Zehnder index ...... 112 5.3 Knots and dynamics ...... 114

6 Number theory, topology and signatures 117 6.1 The modular group SL(2, Z)...... 117 6.2 Torus bundles and their signatures ...... 119 6.3 Lens spaces ...... 123 6.4 Dedekind sums ...... 125

7 Appendix: Algebraic L-theory of rings with involution and the localization exact sequence 128 7.1 Forms over a ring with involution ...... 129 7.2 The localization exact sequence ...... 133

3 Introduction

There is no doubt that quadratic forms are among the most basic and important objects in mathematics. The distinction between an ellipse, a parabola and a hyperbola belongs to the standard curriculum of undergraduate stu- dents in mathematics, and Sylvester’s 1852 Law of Inertia for quadratic forms represents the formalization of this fact. In a suitable basis, any quadratic form in Rn can be written as

2 2 2 2 2 x1 + x2 + ··· + xp − xp+1 − · · · − xp+q for some integers 0 6 p, q 6 n with p + q 6 n. These integers are uniquely defined by the quadratic form, and are the same for two forms if and only if the forms are linearly congruent. The sum p+q is the rank and the difference p−q is the signature of the quadratic form. In other words, Sylvester’s Law of Inertia asserts that for any n > 1 the number of linear congruence classes of n quadratic forms in R is the number of ordered pairs (p, q) with 0 6 p, q 6 n, p + q 6 n (which is equal to (n + 1)(n + 2)/2). It is not a surprise that from these humble beginnings the theory of quadratic forms has percolated to the full body of mathematics, in many different guises. It would require a large encyclopedia to keep track of all these appearances, and it is certainly not our goal to provide such a com- pendium here. In this survey paper, we would like to present some morceaux choisis with the main purpose of serving as a general introduction to the other papers in these volumes, which deal with the signatures of braids. As the title suggests, we shall concentrate on algebra, topology, and dynamics. Even though all the material that we describe is standard, it is usually scattered in the literature. For instance, books and papers dealing with Hamiltonian dynamics usually do not discuss Wall non additivity of signatures. Section 1 contains very standard facts about quadratic forms in Rn. We tried to give some historical flavour to our presentation. A common thread will be Sturm’s theorem, counting real roots of polynomials. From its original formulation as a number of sign variations, we shall describe its reformulation by Sylvester as the signature of a symmetric matrix, and then as an equality in a suitable Witt group. Section 2 is topological. The homology of a manifold is equipped with intersection forms. Therefore signatures came into play, around 1940, and

4 serve as invariants of manifolds. Section 3 deals with the Maslov class. One could easily write thick books on this topic (that probably nobody would read) and we had to limit ourselves to the basics. It is amazing how ubiquitous this class can be and it is sometimes difficult to recognize it. There are connections with mathematical physics, topology, number theory, dynamics, geometry, bounded cohomology etc. In the final sections, we choose three typical applications, among many other possibilities: Section 4 presents some aspects of dynamical systems, specifically hamiltonian, for which signatures are relevant. Section 5 goes deeper in some topological aspects. In particular, we describe the link between the Wall non additivity and the Maslov class. Section 6 describes some fascinating connections with number theory. Section 7 is an appendix on the algebraic L-theory used in surgery obstruction theory, which generalizes some of the algebraic techniques presented here to forms over (noncommutative) rings with involution. We thank the American Institute of Mathematics in Palo Alto, at which we organized a meeting on the “Many facets of Maslov invariants” in April 2014. Here are our signatures:

5 1 Algebra

1.1 The basic deﬁnitions Let us begin with some very basic deﬁnitions from linear algebra. In this section we shall be mainly considering matrices over R. An m × n matrix S = (sij) with entries sij ∈ R (1 6 i 6 m, 1 6 j 6 n) corresponds to a bilinear form (or pairing) of real vector spaces:

m n m n X X S : ((x1, x2, . . . , xm), (y1, y2, . . . , yn)) ∈ R × R 7→ sijxiyj ∈ R. i=1 j=1

∗ ∗ The transpose of an m×n matrix S = (sij) is the n×m matrix S = (sji) ∗ with sji = sij. An n × n matrix S is symmetric if S∗ = S. Quadratic forms correspond to symmetric matrices S. For v = (x1, . . . , xn) ∈ Rn, we set

Q(v) = S(v, v), with S(v1, v2) = (Q(v1 + v2) − Q(v1) − Q(v2))/2.

A quadratic form Q(V ) (or the corresponding symmetric matrix S) is positive deﬁnite (resp. negative deﬁnite) if Q(v) > 0 (resp. Q(v) < 0) for all non zero v ∈ Rn. Two symmetric n × n matrices S,T are linearly congruent if T = A∗SA for an invertible n × n matrix A. A linear congruence can be regarded as an isomorphism A : Rn → Rn of real vector spaces such that

T (v1, v2) = S(Av1, Av2) ∈ R. The spectrum of an n × n matrix S is the set of eigenvalues, i.e. the roots λ ∈ C of the characteristic polynomial det(XIn − S) ∈ R[X]. In general, linearly congruent symmetric matrices S,T do not have the same characteristic polynomial and spectrum. However, it is signiﬁcant (and by no means obvious) that symmetric matrices have real eigenvalues, and that the eigenvalues of linearly congruent symmetric matrices have the same signs. Two n × n matrices S,T are conjugate if T = A−1SA for an invertible n × n matrix A, in which case S,T have the same characteristic polynomial det(XIn − T ) = det(XIn − S) and hence the same eigenvalues. An n × n

6 matrix A is orthogonal if it is invertible and A−1 = A∗. This is equivalent to 2 2 n the preservation of the quadratic form x1 +···+xn in R . Two n×n matrices S,T are orthogonally congruent if T = A∗SA for an orthogonal n × n matrix A, in which case they are both linearly congruent and conjugate.

1.2 Linear congruence

The first basic theorem is that any quadratic form Q(x1, . . . , xn) with real coefficients can be written as a sum, or difference, of squares of linear forms. We refer to [27, Chapter 9] for a description of the complex history of this theorem, with deep roots in the traditional geometry of conic sections and quadrics. Usually, the theorem is attributed to Lagrange who stated it and proved it in full generality in 1759 [82].

J. L. Lagrange (1736–1813)

Theorem 1.1 (18th century). A symmetric matrix is linearly congruent to a diagonal matrix. Proof. Of course, this fact was not expressed in terms of matrices but in terms of quadratic forms: The proof is very easy and is best seen in an example: 2 2 2 2 Q(x1, x2) = x1 + 2x1x2 + 8x2 = (x1 + x2) + 7x2

7 involving completing the square. Formally, the proof goes by induction. If Q 6= 0 there exists some v ∈ Rn such that Q(v) = S(v, v) 6= 0. Then S is S(v, v) 0 linearly congruent to with S0 the (n − 1) × (n − 1)-matrix of 0 S0 S restricted to the (n − 1)-dimensional subspace v⊥ = {w ∈ Rn | S(v, w) = 0} ⊂ Rn. Note that the same proof generalizes to a symmetric matrix over any ﬁeld of characteristic 6= 2. This hypothesis is hidden in the formula S(v1, v2) = (Q(v1 + v2) − Q(v1) − Q(v2))/2. Since any positive real number is a square, one can restate the previous result as: Theorem 1.2. A symmetric matrix is linearly congruent to some   Ip 0 0 Ip,q =  0 0 0  . 0 0 −Iq The crucial observation is that the integers p and q are invariant under linear congruence:

Carl Gustav Jacob Jacobi James Joseph Sylvester (1804-1851) (1814-1897)

8 Theorem 1.3 (Sylvester’s Law of Inertia, 1852-3, [132, 134]). Any symmetric n × n matrix S over R is linearly congruent to a unique matrix of the form Ip,q.

Proof. One has to show that if Ip,q and Ip0,q0 are congruent, one has necessarily p = p0 and q = q0. By contraction, assume for instance that p0 > p. This n 0 would imply that there exists a subspace V+ ⊆ R of dimension p > p with Ip,q(v, v) > 0 for all v ∈ V+\{0}. Such a V+ intersects non trivially any subspace of dimension n − p, and in particular {0} × Rn−p. This implies that there is a vector v ∈ V+\{0} with Ip,q(v, v) 6 0: a contradiction. The original formulation of the Law of Inertia by Sylvester in 1852 [132] was as a “remark ... easily proved” (but for which he did not provide a proof).

A law to which my view of the physical meaning of quantity of matter inclines me, upon the ground of analogy, to give the name of the Law of Inertia for quadratic forms, as expressing the fact of the existence of an invariable number inseparably attached to such forms.

Sylvester’s original 1853 proof [134] of the Law of Inertia is on the next page. Sylvester could not have been aware of an earlier unpublished proof by Jacobi [71]. The proofs by Jacobi and Sylvester are essentially identical. The paper of Jacobi was published in 1857, after his death in 1851, along with a letter by Hermite [60] and a comment by Borchardt [24]. Moreover, Gauss included the Law of Inertia in his 1846 lectures, according to his student Riemann (see [27]). Perhaps, one should speak of the Gauss-Jacobi-Sylvester law.

9 10 1.3 The signature of a symmetric matrix Deﬁnition 1.4. For an n×n symmetric matrix S the integer p in Sylvester’s Law of Inertia 1.3 is called the positive index of S and denoted by τ+(S). Similarly q called the negative index of S and denoted by τ−(S). The signature (also known as the index of inertia) of a symmetric n × n matrix S with entries in R is the diﬀerence

τ(S) = τ+(S) − τ−(S) ∈ {−n, −n + 1,..., −1, 0, 1, . . . , n}.

The signature was only implicit in Sylvester [132]. The terminology was introduced by Frobenius [48]. The rank of S is the sum

n dimR(S(R )) = τ+(S) + τ−(S) ∈ {0, 1, 2, . . . , n}.

Note that S is invertible if and only if τ+(S) + τ−(S) = n. Deﬁnition 1.5.

1. The principal k × k minor of an n × n matrix S = (sij)16i,j6n is

µk(S) = det(Sk) ∈ R with Sk = (sij)16i,j6k the principal k × k submatrix: ! Sk ... S = ......

By convention, we set µ0(S) = 1. 2. An n × n matrix S is regular if µk(S) 6= 0 for 1 6 k 6 n, that is if each Sk is invertible. In particular, Sn = S is invertible. This is a generic assumption, satisﬁed on an open dense set of matrices. Jacobi [70] and Sylvester [132, 133] initiated the expression of the signature of a symmetric matrix as the number of changes of sign in the principal minors.

n+1 Deﬁnition 1.6. The variation of µ = (µ0, µ1, . . . , µn) ∈ R with each µk 6= 0 is

var(µ) = number of changes of sign in µ0, µ1, . . . , µn ∈ {0, 1, 2, . . . , n}.

11 We shall make use of the following two very elementary properties of the variation. Lemma 1.7. 1. The variation of µ is related to the signs of the successive quotients µ1/µ0, µ2/µ1, ... , µn/µn−1 by the identity n X sign(µk/µk−1) = n − 2 var(µ). k=1 2. If the components of µ, µ0 have the same signs except for the kth entry, 0 with sign(µk) = −sign(µk), then  sign(µk/µk−1) + sign(µk+1/µk) if 1 6 k 6 n − 1 0  var(µ ) − var(µ) = sign(µ1/µ0) if k = 0  sign(µn/µn−1) if k = n.

Proof. 1. Consider first the special case n = 1: if µ0, µ1 have the same (resp. different) signs than var(µ0, µ1) = 0 (resp. 1) and the equation is 1 = 1 − 0 (resp. −1 = 1 − 2). For the general case assume inductively true for n and note that sign(µn+1/µn) = 1 − 2 var(µn, µn+1) by the special case. 2. Apply 1. twice. We shall now prove that a regular symmetric matrix S is linearly congruent to the diagonal matrix with entries µk(S)/µk−1(S) (1 6 k 6 n). The proof is by the “algebraic plumbing” of matrices – the algebraic analogue of the geometric plumbing of manifolds, discussed in section 2 below. Definition 1.8. The plumbing of a regular symmetric n × n matrix S with respect to v ∈ Rn, w 6= vS−1v∗ ∈ R is the regular symmetric (n+1)×(n+1) matrix S v∗ S0 = . v w The Sylvester-Jacobi-Gundelfinger-Frobenius theorem gives a very simple way of calculating the signature, at least in the case of a regular matrix. Theorem 1.9 (Sylvester [132], Jacobi [70], Gundelfinger [57], Frobenius [48]). The signature of a regular symmetric n × n matrix S is given by: n P τ(S) = sign(µk(S)/µk−1(S)) k=1 = n − 2 var(µ0(S), µ1(S), . . . , µn(S)) ∈ {−n, −n + 1, . . . , n}.

12 Proof. Since every regular symmetric n × n matrix S is obtained from the 0 × 0 matrix by n successive plumbings, it suﬃces to calculate the jump in signature under one plumbing. The matrix identity

S v∗ 1 0 S 0 1 S−1v∗ S0 = = v w vS−1 1 0 w − vS−1v∗ 0 1 shows that the symmetric (n + 1) × (n + 1) matrix S0 is linearly congruent S 0 to with 0 w − vS−1v∗

0 0 −1 ∗ µk(S ) = µk(S) (1 6 k 6 n), µn+1(S ) = µn(S)(w − vS v ).

Another way of expressing the same result is the following. The positive index of S is the number of indices k = 1, . . . , n such that µk−1(S) and µk(S) have the same signs

τ+(S) = p = n − var(µ0(S), µ1(S), . . . , µn(S)), the negative index of S is the number of indices for which the signs are diﬀerent τ−(S) = q = var(µ0(S), µ1(S), . . . , µn(S)). and the signature is the diﬀerence

τ(S) = τ+(S) − τ−(S) = p − q = n − 2 var(µ0(S), µ1(S), . . . , µn(S)).

Corollary 1.10. If S is an invertible symmetric n × n matrix which is not regular then for suﬃciently small 6= 0 the symmetric n × n matrix S = S + In is regular, and

n P τ(S) = τ(S) = sign(µk(S)/µk−1(S)) ∈ {−n, −n − 1, . . . , n − 1, n}. k=1

1.4 Orthogonal congruence The next step in the understanding of symmetric matrices concerns orthogonal congruence rather than linear congruence.

13 Augustin Cauchy (1789–1857)

Theorem 1.11 (Spectral theorem, Cauchy 1829, [34]). 1. Any symmetric matrix S with real coeﬃcients is orthogonally congruent to a diagonal matrix. The eigenvalues of S are real numbers. 2. Two symmetric n × n matrices S,T are orthogonally congruent if and only if their eigenvalues are the same.

There are many proofs of this truly fundamental fact. We shall hint at nine proofs! For a history of this result, we recommend Hawkins [59]. It is interesting to compare this spectral theorem with theorem 1.1. The classification of real quadratic forms under congruence is elementary and its proof extends to any field with characteristic different from 2. In the case of R, there is a finite number of normal forms for each dimension n. On the other hand, the spectral theorem is specific to the field of real numbers and involves an infinite number of normal forms since the eigenvalues are invariant under conjugacy. Historically, the subtlety between these two important theorems has not always been clear to the founding fathers of linear algebra. An immediate corollary of the Law of Inertia and the spectral theorem is that the signs of the eigenvalues of a symmetric matrix are invariant under congruence.

Corollary 1.12 (of Sylvester’s Law of Inertia 1.3).

14 1. Two symmetric matrices S,T are linearly congruent if and only if their eigenvalues have the same signs. 2. The positive (resp. negative) index τ+(S) (resp. τ−(S)) of a symmetric n × n matrix S is the number of eigenvalues λk > 0 (resp. λk < 0). The signature of S is sum of the signs of the eigenvalues

n X τ(S) = sign(λk) ∈ {−n, −n + 1, . . . , n − 1, n}. k=1 We shall now describe many proofs of the spectral theorem, following more or less the chronological order. Each of these proofs casts some light on the spectral theorem. We denote by h , i the usual scalar product on Rn. Note ﬁrst that the orthogonal of any S-invariant subspace is also S-invariant. Observe also that two eigenvectors v1, v2 with diﬀerent real eigenvalues λ1, λ2 have to be orthogonal for h , i. Indeed

hS(v1), v2i = hv1,S(v2)i = λ1hv1, v2i = λ2hv1, v2i. so that hv1, v2i = 0. In other words, eigenspaces are orthogonal so that the point of the The- orem is to prove that the spectrum is real.

The heuristic first “proof” is physical and half convincing for a contemporary mathematician but it was considered solid by Lagrange in 1762 [83]. Let us explain it in modern language. Suppose first that the quadratic form 1 U(v) = 2 hS(v), vi is definite positive and consider this function as a mechanical potential, producing a force −grad U(v) = −S(v). Assuming a unit mass, the equation of motion is

d2v = −S(v). dt2 Conservation of energy yields

1 dv 2 + U(v) = Constant 2 dt and this implies stability: any trajectory stays in a bounded neighborhood of the origin since U(v) remains bounded. Recall that the diﬀerential equation

15 d2v/dt2 + ω2v = 0 has solutions v = a exp(±iωt). These solutions can only be bounded (as the time runs over R) if ω is a real number. Indeed exp(i(α+ iβ)t) = exp(iαt) exp(−bt) is bounded if an only if b = 0. “Therefore”, S has only real (and positive) eigenvalues. If S is not deﬁnite positive, just consider S + λId with a large positive real number λ.

Au reste, quoiqu’il soit difficile, peut-être impossible, de déterminer en général les racines de l’équation P = 0, on peut cependant s’assurer, par la nature mêmedu problème,que ces racines sont nécessairement toutes réelles [...]; car sans cela les valeurs de y0, y00, y000,... pourraient croˆıtre àl’infini, ce qui serait absurde [83].

Cauchy’s proof (1829) [34]. Let λ be a complex eigenvalue of S and let v be a non zero eigenvector in Cn. Of course,v ¯ is an eigenvector with eigenvalue λ¯. We still denote by the same symbol h , i the extension to Cn of the scalar product as a bilinear form (not as a hermitian form). We have

hS(v), v¯i = λhv, v¯i = hv, S(¯vi = λ¯hv, v¯i ∈ C. Since v 6= 0, we have hv, v¯i= 6 0. This implies that λ = λ¯ so that λ is indeed real. Amazingly, this proof, which looks crystal clear today, was not so convincing at the beginning of the nineteenth century, probably because complex numbers still sounded mysterious (they were called imaginary or even impossible).

An algebraic proof. One observes ﬁrst that the theorem is obvious in dimension 1 and very easy in dimension 2. Indeed, if n = 2, one can for example subtract from S a suitable multiple of the identity matrix so that one can assume that a b S = b −a √ which has ± a2 + b2 as eigenvalues. Now, if n > 3, one uses the fact that any matrix with real entries has an invariant 2 dimensional subspace E. Of course, since S is symmetric, the orthogonal of E is also invariant and the proof goes by induction. One could argue that the standard proof of the existence of an invariant 2 dimensional subspace uses complex numbers so that this proof is very close to Cauchy’s proof.

16 Just for fun, let us describe a magic proof given by Sylvester (1852) [132].

Consider the characteristic polynomial P (X) = det(XIn − S). Note that 2 2 P (X)P (−X) = det(X In − S ). Since S is symmetric, all diagonal entries of S2 are sums of squares and are therefore non negative. Let us write

2 n 2 n−1 2 n−2 P (X)P (−X) = (−X ) + a1(−X ) + a2(−X ) + ... + an.

2 The ﬁrst coeﬃcient a1 is the trace of S , the sum of the squares of the entries of S, and is therefore non negative. More generally ak is the sum of squares of the k × k minors of S and is therefore non negative as well. Sylvester concludes that this implies that P cannot have an imaginary non real root. Assuming some root of the form α+iβ, with a, b real, one could replace S by S − αIn and reduce the problem to proving that a purely imaginary root iβ is not possible. Now, that would imply that the positive number β2 would n n−1 n−2 be a root of X +a1X +a2X +...+an = 0 which is clearly impossible. Sylvester describes his proof as “very simple”. He was aware of the fact that Cauchy had published a proof “somewhere” (in his words) but one can understand why he preferred his own proof. Indeed this approach is truly algebraic and does not allow any intrusion into the complex numbers. It only uses the fact that a sum of squares is positive and that a positive number is not zero.

The classical topological proof (ﬁrst used implicitly by Euler in dimension 3). One considers a vector v that maximizes the value of S(v, v) on the unit sphere of Rn. Note that S(v +tw, v +tw) = S(v, v)+2tS(v, w)+t2S(w, w) so that one has S(v, w) = 0 for every vector w orthogonal to v. In other words, v is an eigenvector for the matrix S. One also concludes by induction.

Let Q be quadratic form on some n-dimensional real vector space E equipped with a positive deﬁnite scalar product h, i and let S be the symmetric endomorphism associated to Q, i.e. such that Q(v) = hS(v), vi. Denote by λ1(S) 6 ··· 6 λn(S) the n real eigenvalues of S (whose existence follows

17 from the spectral theorem). Clearly, λn(S) is the maximum value of the quotient Q(v)/hv, vi as v describes E \{0}. More generally, the Rayleigh minimax theorem asserts that

λk(S) = min max{Q(v)/hv, vi | v ∈ F ; v 6= 0; dim(F ) = k}. F v

Actually, one can get a more pedantic (sixth) proof of the spectral theorem using this minimax idea and some modern algebraic topology. This proof is not by induction. Note that if S is invertible, the function S(v, v)/ < v, v > is constant on lines and therefore deﬁnes a function on the real projective n−1 space RP . A knowledgeable reader knows that a function on a compact manifold must have at least as many critical points as the sum of the Betti numbers (for instance modulo 2). In this case, the sum is equal to n and each of these critical points yields an eigenvector.

Note that a corollary of Rayleigh minimax is that the eigenvalues of any (n − 1) × (n − 1) minor of S are interlaced with the n eigenvalues λk(S). One can give a simple algebraic proof of this fact, which will provide our seventh proof of the spectral theorem (from [18]). The proof goes by induction. Assume that all (n−1)×(n−1) symmetric matrices are diagonalizable in an orthonormal basis. Let S be an n × n symmetric matrix. We can therefore assume it has the following form in some orthonormal basis.   λ1 0 0 ... 0 a1  0 λ2 0 ... 0 a2     0 0 λ ... 0 a   3 3  S =  ......   ......     0 0 0 . . . λn−1 an−1 a1 a2 a3 . . . an−1 λn with λ1 6 λ2 6 ··· 6 λn−1. Let us evaluate the characteristic polynomial P (X) = det(XIn − S) of S.

n−1 n X 2 Y Y P (X) = − ai (X − λj) + (X − λi). i=1 j6=i;16j6n−1 i=1

Let us ﬁrst make the generic assumption that all the λi are distinct and that the ai’s are not zero. It follows that the signs of P (−∞), P (λ1),...,P (λn−1),

18 P (+∞) alternate. Hence there exist n real roots in the corresponding intervals so that S is indeed diagonalizable. The general case follows since the set of real polynomials having all their roots in R is closed. As a bonus, we provide a eighth proof, from Appell (1925) [5, Chapter 1], even though it is probably neither the best nor the easiest! Let S = (sij) be a real symmetric n × nmatrix. As before we denote by µk(S) = det(Sk) the principal minors. Denote by C = (cij) the comatrix of S so that cij is (−1)i+j times the determinant of the (n − 1) × (n − 1) matrix obtained from S by deleting the ith row and the jth column. We claim that

cn−1,n−1 cn−1,n = µn(S)µn−2(S). cn,n−1 cn,n This follows from the computation of the product of the two determinants

1 0 ... 0 0 0

0 1 ... 0 0 0

......

...... cn−1,n−1 cn−1,n ...... = cn,n−1 cn,n 0 0 ... 1 0 0

cn−1,1 cn−1,2 . . . cn−1,n−2 cn−1,n−1 cn−1,n

cn,1 cn,2 . . . cn,n−2 cn,n−1 cn,n and s1,1 s1,2 . . . s1,n−2 s1,n−1 s1,n

s2,1 s2,2 . . . s2,n−2 s2,n−1 s2,n

......

...... = µn(S)

sn−2,1 sn−2,2 . . . sn−2,n−2 sn−2,n−1 sn−2,n

sn−1,1 sn−1,2 . . . sn−1,n−2 sn−1,n−1 sn−1,n

sn,1 sn,2 . . . sn,n−2 sn,n−1 sn,n and the observation that C.S = (det S)In = µn(S)In. Let us make the generic assumption that the polynomials µk(XIn − S)16k6n have distinct roots and they are all distinct. Let us prove by induction on k that µk(XIn −S) has k real roots, and that there is a unique root of µk−1(XIn − S) between two roots of µk(XIn − S). Assume this is true up to k = n − 1 and denote by α1 < ··· < αn−1 the real roots of µn−1(XIn − S).

19 Apply the previous identity to the symmetric matrix αiIn − S (so that µn−1(αiIn − S) = 0). We get that µn(αiIn − S)µn−2(αiIn − S) is negative. Hence there is at least one root of µn(XIn − S) in each interval [αi, αi+1]. A degree consideration also shows that there is also at least a root < α1 and another root > αn−1 and this concludes the inductive step. One should still get rid of the generic assumption on the roots of µk(XIn−

S)16k6n. As in the sixth proof, this follows from the fact that the set of real polynomials having all their roots in R is closed. The ninth proof will be explained later (see 1.6).

The spectral theorem expresses the fact that there is a basis in Rn which is orthonormal for both h, i and S. A more invariant way of expressing the same theorem is the following. Let S,T be two quadratic forms on a real vector space. If T is definite positive, then there exists a basis in which the matrix of T is the identity (i.e. orthonormal for T ) and the matrix of S is diagonal. There is a nice generalization of the previous statement due to Milnor (cf. Greub [54, Chapter IX.3]). Let S,T be two quadratic forms on a real vector space Rn of dimension at least 3. Assume that there is no vector v which is simultaneously isotropic for S and T , i.e. such that S(v, v) = T (v, v) = 0. Then there is a basis in which both S and T are diagonal. The proof is the following. Consider the map from the unit sphere Sn−1 to n−1 C\{0} sending v to S(v, v)+iT (v, v). Since n > 3, the sphere S is simply connected so that there is a continuous determination of the argument of S(v, v) + iT (v, v) as a continuous function on Sn−1 with values in R. Hence, there must exist a point on Sn−1 for which this argument achieves it maximum. One shows easily that such a point v is a simultaneous eigenvector for S and T . One then proceeds by induction on n. Even though this is not the purpose of this paper, one should remember that one of the most important motivations for the spectral theorem comes from mechanics. In the simplest situation, if a rigid body rotates around a fixed point, with some angular velocity Ω ∈ R3, its kinetic energy is a (positive definite) quadratic form in Ω. An orthonormal basis in which this form is diagonal defines the three principal inertial axes and the eigenvalues are the principal inertial moments. For more information, see Arnold [9, Chapter 6].

20 This spectral theorem had not only a profound impact on mathematics but also on the Moon! Lagrange, Cauchy, Jacobi and Sylvester have their craters on the Moon.

Lagrange Cauchy

Jacobi Sylvester

21 1.5 Continued fractions, tridiagonal matrices and signatures In 1853 Sylvester [133] established a link between the 1829 theorem of Sturm on the number of real roots of a real polynomial, the signature of a tridiagonal matrix and improper continued fractions. We shall discuss the link in section 1.6 below. n The proper continued fraction of χ = (χ1, χ2, . . . , χn) ∈ R is the real number 1 χ + ∈ 1 1 R χ2 + .. χ3 + . 1 + χn assuming there are no divisions by 0. Continued fractions have a long and distinguished history – we refer to Karpenkov [73] for a recent account.

Example 1.13. Given integers p0 > p1 > 1 let q1, q2, . . . , qn > 1 be the successive quotients and p2, . . . , pn > 0 the successive remainders in the iterations of the Euclidean algorithm

pk+1 = pkqk − pk−1 ∈ Z (1 6 k 6 n) , 0 6 pk+1 < pk, qk = bpk−1/pkc, pn = greatest common divisor(p0, p1), pn+1 = 0.

Then 1 p /p = q + ∈ . 0 1 1 1 Q q2 + .. q3 + . 1 + qn In terms of matrices, the recurrence is

p 0 1 0 1 0 1 p k = ... 0 . pk+1 −1 qk −1 qk−1 −1 q1 p1

However, in dealing with signatures we need to be more concerned with improper continued fractions, following Sylvester. The improper continued

22 n fraction of χ = (χ1, χ2, . . . , χn) ∈ R is the real number 1 [χ , χ , . . . , χ ] = χ − ∈ 1 2 n 1 1 R χ2 − .. χ3 − . 1 − χn assuming there are no divisions by 0.

n n+1 Lemma 1.14. Let χ = (χ1, . . . , χn) ∈ R , µ = (µ0, µ1, µ2, . . . , µn) ∈ R be vectors satisfying the recurrence

µk = µk−1χk − µk−2 (1 6 k 6 n) with µ−1 = 0, µ0 6= 0 ∈ R.

The continued fractions [χk, χk−1, . . . , χ1] are well-deﬁned if and only if µk 6= 0 ∈ R, in which case there are identities

µk/µk−1 = [χk, χk−1, . . . , χ1] ∈ R (1 6 k 6 n). Proof. By induction on n, using the identities

µ1/µ0 = χ1, µ2/µ0 = χ2χ1 − 1 = [χ2, χ1][χ1], 1 + [χk, χk−1, . . . , χ1][χk−1, χk−2, . . . , χ1] = [χk−1, χk−2, . . . , χ1]χk (2 6 k 6 n).

In terms of matrices, the recurrence is

µ χ −1 χ −1 χ −1 µ k = k k−1 ... 1 0 . µk−1 1 0 1 0 1 0 0

Example 1.15. Given integers p0, p1 ∈ Z\{0} let q1, q2, . . . , qn ∈ Z\{0} be the successive quotients and p2, . . . , pn ∈ Z\{0} the successive remainders in the iterations of the Euclidean algorithm (up to sign)

pk+1 = pkqk − pk−1 (1 6 k 6 n), 0 6 |pk+1| < |pk| |pn| = greatest common divisor(|p0|, |p1|), pn+1 = 0. In terms of matrices, the recurrence is

p q −1 q −1 q −1 p k+1 = k k−1 ... 1 1 pk 1 0 1 0 1 0 p0

23 which can be inverted to read p q −1 q −1 q −1 p n−k−1 = n−k n−k+1 ... n n . pn−k 1 0 1 0 1 0 0

The vectors

n χ = (χ1, χ2, . . . , χn) = (qn, qn−1, . . . , q1) ∈ Z , n+1 µ = (µ0, µ1, . . . , µn) = (pn, pn−1, . . . , p0) ∈ Z satisfy the hypothesis of Lemma 1.14, so that

f = pk/pk+1 = [qk+1, qk+2, . . . , qn] ∈ Q (0 6 k 6 n − 1).

Deﬁnition 1.16. The tridiagonal symmetric n×n matrix of χ = (χ1, χ2, . . . , χn) ∈ n is R   χ1 1 0 ... 0 0  1 χ2 1 ... 0 0     0 1 χ ... 0 0   3  Tri(χ) =  ......  .  ......     0 0 0 . . . χn−1 1  0 0 0 ... 1 χn

Tridiagonal matrices were studied in particular by Jacobi [69]. They are ubiquitous in mathematics, in recurrences, Sturm theory, numerical analysis, orthogonal polynomials, integrable systems and in the Hirzebruch-Jung resolution of singularities.

n Deﬁnition 1.17. A vector χ = (χ1, χ2, . . . , χn) ∈ R is regular if Tri(χ) is regular in the sense of Deﬁnition 1.5, that is if µk(Tri(χ)) 6= 0 ∈ R (1 6 k 6 n).

Proposition 1.18. n 1. For any χ = (χ1, χ2, . . . , χn) ∈ R the principal minors µk = µk(Tri(χ)) ∈ R satisfy the recurrence of Lemma 1.14

µk−2 + µk = χkµk−1 (1 6 k 6 n) with µ−1 = 0, µ0 = 1.

24 2. χ is regular if and only if

χk 6= µk−2/µk−1 ∈ R (1 6 k 6 n), if and only if each [χk, χk−1, . . . , χ1] (1 6 k 6 n) is well-deﬁned. If these conditions are satisﬁed

µk/µk−1 = [χk, χk−1, . . . , χ1] 6= 0 ∈ R (1 6 k 6 n). n+1 n 3. For any (p0, p1, . . . , pn) ∈ (R\{0}) , (q1, q2, . . . , qn) ∈ (R\{0}) satisfying pk+1 = pkqk − pk−1 (1 6 k 6 n), pn+1 = 0 the vectors n χ = (χ1, χ2, . . . , χn) = (qn, qn−1, . . . , q1) ∈ R , n+1 µ = (µ0, µ1, . . . , µn) = (pn, pn−1, . . . , p0) ∈ R satisfy the recurrences of 1., with Tri(χ) regular and µk = µk(Tri(χ)) such that

µn−k/µn−k−1 = [χn−k, χn−k−1, . . . , χ1] = pk/pk+1 = [qk+1, qk+2, . . . , qn] ∈ R (0 6 k 6 n − 1). Theorem 1.19 (Sylvester 1853 [133]). The signature of a regular tridiagonal symmetric n × n matrix Tri(χ) is given by n n P P τ(Tri(χ)) = sign([χk, χk−1, . . . , χ1]) = sign(µk/µk−1) k=1 k=1 = n − 2 var(µ0, µ1, . . . , µn) ∈ {−n, −n + 1, . . . , n} with µk = µk(Tri(χ)), µk/µk−1 = [χk, χk−1, . . . , χ1] 6= 0 ∈ R. Proof. Apply the Sylvester Jacobi Gundelﬁnger Frobenius Theorem 1.9 Lemma 1.14 and Proposition 1.18 2., noting that the eigenvalues of Tri(χ) have the signs of µk/µk−1 (1 6 k 6 n). Remark 1.20. The right hand side of Sylvester’s formula is not clearly invari- n ant under permutation of the indices. For any χ = (χ1, χ2, . . . , χn) ∈ R and any permutation σ of {1, . . . , n} let n χσ = (χσ(1), χσ(2), . . . , χσ(n)) ∈ R .

The symmetric n × n matrices Tri(χ), Tri(χσ) are orthogonally congruent, via the orthogonal n × n matrix P (σ) obtained from the identity In by per- muting the columns according to σ. Hence the eigenvalues are the same, and τ(Tri(χ)) = τ(Tri(χσ)).

25 Sylvester found this invariance remarkable and even worked out a proof “by hand” when n = 5. We leave the reader with this challenging exercise. The key to Sylvester’s application of Theorem 1.19 to Sturm’s theorem was the observation of this invariance of τ(Tri(χ)) on the ordering of the χk’s, in particular under the permutation (χ1, χ2, . . . , χn) → (χn, χn−1, . . . , χ1).

Extract from Sylvester [133]

As an artist delights in recalling the particular time and atmo- spheric eﬀects under which he has composed a favourite sketch, so I hope to be excused putting upon record that it was listening to one of the magniﬁcent choruses in the “Israel in Egypt” that, unsought and unsolicited, like a ray of light, silently stole into my mind the idea (simple, but previously unperceived) to the equivalence of the Sturmian residues to the denominator series formed by the inverse convergents. The idea was just what was wanting, – the key-note to the due and perfect evolution of the theory.

26 1.6 Sturm’s theorem and its reformulation by Sylvester

Jacques Charles Fran¸cois Sturm, ForMemRS (1803-1855) on “his” asteroid “Sturm 31043”, almost 4 million km from the Sun (Artist’s impression)

Many contemporary mathematicians have forgotten the numerical aspect of theoretical algebra. If a polynomial with real coefficients is given, how can one determine in practice its roots with a given accuracy? Modern comput- ers give us the feeling that it suffices to type the command “Solve P (x) = 0” to get an immediate answer. As a matter of fact, treatises on Algebra, at least until the end of the nineteenth century, gave a strong emphasis on this numerical problem. For instance, the classical “Cours d’algèbresupérieure”, by J.A. Serret [125], dated 1877, is one of the first textbooks with a thorough presentation of Galois theory. It contains several chapters on the numerical aspect and splits the topics in two distinct problems. The first is the separation of roots: one has to count the number of roots in a given interval, in order to locate intervals containing a single root. The second consists of var- ious numerical methods enabling to shrink such an interval to any desirable length. Concerning the separation problem, there is no doubt that the most impressive theorem is due to Sturm.

27 L’algèbre offrait une lacune regrettable, mais cette lacune se trouva combléede la manière la plus heureuse par le fameux théorème de Sturm. Ce grand géomètre communiqua àl’Académiedes Sciences, en 1829, la démonstration de son théorèmequi con- stitue l’une des plus brillantes découvertes dont se soit enrichie l’Analyse mathématique.

Amazingly, one of the “most brilliant discoveries in Analysis” is only familiar today to a very tiny minority of mathematicians.

Deﬁnition 1.21. Let P (X) ∈ R[X] be a degree n polynomial. 1. The Sturm functions of P (X) are the sequences of polynomials

(P0(X),P1(X),...,Pn(X)), (Q1(X),Q2(X),...,Qn(X))

occurring as the successive remainders and quotients in the iterations of the Euclidean algorithm with

0 P0(X) = P (X),P1(X) = P (X), Pk+1(X) = Pk(X)Qk(X) − Pk−1(X) (1 6 k 6 n), deg(Pk+1(X)) < deg(Pk(X)), Pn(X) = constant,Pn+1(X) = 0.

2. The polynomial P (X) is regular if the real roots of the polynomials P0(X),P1(X),...,Pn(X) are all distinct and non-zero, with Pn(X) ∈ R ⊂ R[X] constant. In particular P (X) has no repeated roots, deg(Pk(X)) = n−k, and deg(Qk(X)) = 1. Call a ∈ R regular if each of P0(a),P1(a),..., Pn(a) ∈ R is non-zero, in which case the variation of a is deﬁned to be the number of sign changes in this sequence

var(a) = var(P0(a),P1(a),...,Pn(a)) ∈ {0, 1, . . . , n}

(cf. Deﬁnition 1.6).

Theorem 1.22 (Sturm 1829, [130]). For a regular degree n polynomial P (X) ∈ R[X] and regular a, b ∈ R with a < b the number of real roots of P (X) ∈ R[X] contained in [a, b] is equal to var(a) − var(b) ∈ {0, 1, . . . , n}.

28 Proof. The proof is clever, but not diﬃcult. The key point is that

Pk+1(X) + Pk−1(X) = Pk(X)Qk(X) so that for every real root x of Pk(X), the signs of Pk+1(x) 6= 0 and Pk−1(x) 6= 0 are different (1 6 k 6 n − 1). Consider var(x) as a function of x ∈ [a, b]. Strictly speaking, it is only defined on the set U ⊂ [a, b] of regular x which is the complement of the finite set of real roots of some Pk(X) (0 6 k 6 n − 1). On each connected component of U, var is obviously constant. We therefore have to describe the jumps of var(x) when x crosses a real root x0 of some Pk(X), with 0 6 k 6 n − 1. Let us compare var(x0 − ) and var(x0 + ) for > 0 small enough. When x goes from x0 − to x0 + , all components of

(P0(x),P1(x),...,Pn(x)) keep the same sign, except the one corresponding to the polynomial Pk for which Pk(x0) = 0. If 1 6 k 6 n−1, the signs of the two adjacent components, Pk−1(x),Pk+1(x) are diﬀerent. Therefore, independently of the sign of Pk(x ± ), we have var(Pk−1(x ± ),Pk(x ± ),Pk+1(x ± ) = 1. In particular var(x0 − ) = var(x0 + ) so that var(x) is constant when x crosses x0. Therefore there can only be a jump in var(x) when x crosses a root x0 of P = P0. There are two possibilities: either P is increasing at the 0 real root x0, in which case f(x1) > 0 and P (x1) > 0, or P is decreasing 0 at the root x0, in which case f(x1) < 0 and P (x1) < 0. In both cases var(x0 − ) − var(x0 + ) = 1. We have stated Sturm’s theorem in the generic case of a regular polynomial. We leave to the reader the formulation and proof of the general case, including possible multiple roots. Let us work out two simple examples. For c ∈ R let P (X) = X + b ∈ R[X], so that

P0(X) = X + b, P1(X) = 1,Q1(X) = X + b.

Then P (X) is regular, and any a 6= −b ∈ R is regular, with ( 0 if x > −b var(x) = var(x + b, 1) = 1 if a < −b.

29 As x ∈ R jumps across the real root −b (from below) of P (X) the variation var(x) decreases by 1. For b, c ∈ R let P (X) = X2 + bX + c ∈ R[X], so that

2 2 P0(X) = X + bX + c, P1(X) = 2X + b, P2(X) = b /4 − c, 2 Q1(X) = X/2 + 1/4,Q2(X) = (2X + b)/(b /4 − c).

2 Assuming that b /4 − c < 0 we have that P0(X) has no real roots, is regular, and that any x ∈ R such that 2x + b 6= 0 is regular, with var(x) = var(x2 + bx + c, 2x + b, b2/4 − c) = 1.

As x ∈ R jumps across the real root −b/2 of P1(X) the variation var(x) remains constant. In 1853 Sylvester [133] established a link between continued fractions, signatures of tridiagonal matrices and Sturm’s theorem. He noticed that one can use the dictionary between continuous fractions and tridiagonal matrices in the context of polynomials instead of integers. This enabled him to recast Sturm’s theorem in terms of signatures of tridiagonal matrices whose entries are polynomials in one real variable. The main point is to transform Sturm’s theorem into a purely algebraic fact, quite independent of the topology of the ﬁeld of real numbers. Starting from a polynomial, one constructs a canonical symmetric matrix whose signature contains the relevant information on the number of real roots. Theorem 1.23 (Sylvester 1853). For any regular degree n polynomial P (X) ∈ R[X] with Sturm functions (P0(X),P1(X),...,Pn(X)), (Q1(X),Q2(X),..., Qn(X)) deﬁne the symmetric tridiagonal n × n matrix with entries in R[X]   Q1(X) 1 0 ... 0 0  1 Q2(X) 1 ... 0 0     0 1 Q (X) ... 0 0   3  Tri(Q)(X) =  ......  .  ......     0 0 0 ...Qn−1(X) 1  0 0 0 ... 1 Qn(X)

For any regular a ∈ R the signature of the symmetric tridiagonal n×n matrix Tri(Q(a)) with entries in R is τ(Tri(Q)(a)) = n − 2 var(a) ∈ {−n, −n + 1, . . . , n}.

30 Proof. Applying Proposition 1.18 3. with

χ = (χ1, . . . , χn) = (qn, . . . , q1) = (Qn(a),...,Q1(a)), µ = (µ0, . . . , µn) = (pn, . . . , p0) = (Pn(a),...,P0(a)) we have that Tri(χ) is regular, with µk = µk(Tri(χ)) such that

µn−k/µn−k−1 = [χn−k, . . . , χ1] = [qk+1, . . . , qn] = [Qk+1(a),...,Qn(a)] = pk/pk+1 = fk(a)/fk+1(a) ∈ R (0 6 k 6 n − 1).

Moreover, Tri(Q(a)) = Tri(χ)σ with σ(k) = n − k + 1, so

τ(Tri(Q(a))) = τ(Tri(χ)) (by Remark 1.20) n P = sign(µk/µk−1) (by Theorem 1.19) k=1 n n−1 P P = sign(pn−k/pn−k+1) = sign(pk/pk+1) (by reordering) k=1 k=0 = n − 2 var(p0, p1, . . . , pn) (by Lemma 1.7 1.) = n − 2 var(a) ∈ {−n, −n + 1, . . . , n}.

Corollary 1.24 (of Sturm’s Theorem 1.22 and Sylvester’s Theorem 1.23). For any regular degree n polynomial P (X) and regular a, b ∈ R with a < b the number of roots in the interval [a, b] is

var(a) − var(b) = (τ(Tri(Q(b)) − τ(Tri(Q(a)))/2.

Proof. The number of real roots of P (X) ∈ R[X] contained in [a, b] is equal to var(a) − var(b) = (n − τ(Tri(Q(a)))/2 − (n − τ(Tri(Q(b)))/2 = (τ(Tri(Q(b)) − τ(Tri(Q(a)))/2 ∈ {0, 1, . . . , n}.

The book Barge and Lannes [17] is a far-reaching generalization of these remarks, oﬀering algebraic connections between Sturm sequences, the signatures of tridiagonal matrices and Bott periodicity. Sylvester wrote magisterially of the relationship between algebra and geometry (which we do not wholly share):

31 Aspiring to these wide generalizations, the analysis of quadratic functions soars to a pitch from whence it may look proudly down on the feeble and vain attempts of geometry proper to rise to its level or to emulate it in its ﬂights. (1850)

J.J.Sylvester Savilian Professor of Geometry, Oxford, 1883-1894

There are many other ways of connecting the number of roots of polynomials to the signature of symmetric matrices. We shall mention only two of them. A polynomial P (X) ∈ R[X] of degree n with simple roots gives rise to an n-dimensional R-algebra AP (X) of dimension n, namely R[X]/P (X). The structure of this algebra is clear: it is the direct sum of as many copies of R as there are real roots and of as many copies of C as pairs of complex conjugate pairs of roots. The multiplication by an element u in AP (X) deﬁnes a linear endomorphism u of AP (X). Therefore, AP (X) is canonically equipped with a 2 quadratic form φP (X) given by φP (X)(u) = trace(u ). Clearly, any real root gives rise to a + sign in the signature of φP (X) and every pair of complex roots gives a (+, −) contribution. Hence the signature of φP (X) is number of real roots. To compute explicitly the signature of φP (X), one can use the natural 2 n−1 basis of AP (X) given by 1,X,X , ..., X . In this basis, the matrix of multiplication by X in AP (X) is the companion matrix C(P ) of P and its trace is

32 the sum σ1 of the roots of P . In the same way the trace of the multiplication k k by X is the trace of C(P ) , i.e. the sum σk of the k-th powers of the roots of P . n n−1 Speciﬁcally, if P (X) = X − an−1X − · · · − a0, the companion matrix is   0 0 0 ... 0 a0 1 0 0 ... 0 a1    0 1 0 ... 0 a   2  C(P ) = ......  ......    0 0 0 ... 0 an−2  0 0 0 ... 1 an−1, with characteristic polynomial det(XIn − C(P )) = P (X). The matrix of the i j i+j form AP (X) in this basis is given by φP (X)(X ,X ) = trace(C(P ) ) = σi+j. We therefore get the following theorem.

Theorem 1.25. If P (X) is a polynomial in R[X] with distinct roots, the number of its real roots is equal to the signature of the symmetric n × n Hermite matrix   σ0 σ1 σ2 . . . σn−2 σn−1  σ1 σ2 σ3 . . . σn−1 σn     σ σ σ . . . σ σ   2 3 4 n n+1  S =  ......   ......    σn−2 σn−1 σn . . . σ2n−4 σ2n−3 σn−1 σn σn+1 . . . σ2n−1 σ2n−2 where σk denotes the sum of the k-th powers of all the roots of P . By Theorem 1.9 the symmetric matrix S has signature n if and only if k µk(S) > 0 for k = 1, 2, . . . , n. In view of the identities σk = trace(C(P ) ) this is an explicit set of n inequalities in the coeﬃcients of the polynomial P (X) which are satisﬁed if and only if all the roots of P (X) are real. As a very simple example let us compute the Hermite matrix for a degree two polynomial X2 + bX + c. The companion matrix is

0 −c C = 1 −b

33 so that trace(C0) = 2, trace(C) = −b and trace(C2) = b2 − 2c. The corresponding Hermite matrix is therefore

2 −b −b b2 − 2c whose signature is 0 if b2 −4c < 0 and 2 if b2 −4c > 0, confirming high school algebra! The story of the previous theorem is rather tortuous. In a first step, Sylvester tried to understand each polynomial in the Sturm sequence Pi of a given polynomial P = P0, directly as a function of the roots of P . His paper, dated 1839 and published in 1841, contains nice formulas but no proofs [131]. Proofs were provided by Sturm himself in 1841. In 1847, Borchardt reformulated this Sylvester-Sturm theorem, showing the link with the σk [23]. The modern proof that we provided is certainly not original and can be found for example in Gantmacher [52]. 2 The n -dimensional R-algebra Mn(R) of n × n matrices is equipped with a canonical quadratic form: Φ(M) = trace(M 2). Notice that the trace of the square of a symmetric matrix is the sum of squares of all coefficients, and is therefore positive definite. The restriction of Φ(M) to any subspace of symmetric matrices is therefore also a positive definite quadratic form. Let S be a symmetric matrix in R and let P (X) = det(XIn − S) ∈ R[X] be its characteristic polynomial. Assume that all roots of P (X) are distinct. The Cayley-Hamilton theorem implies that the algebra AP (X) = R[X]/P (X) embeds as an n-dimensional subspace in Mn(R), sending X to S. As S is symmetric, the quadratic form φP (X) is the restriction of Φ to this subspace, so we conclude that φP (X) is positive definite, and Theorem 1.25 gives that all roots of P (X) are real. This is the ninth proof of the spectral theorem, due to Borchardt [23]. According to Serret, this proof is “la plus remarquable” of the spectral theorem [125]. Given a polynomial P (X), we know that the fact that all its roots are real can be expressed by the positivity of n polynomials in the coefficients of P (X). Borchardt’s proof shows more: that these polynomials are actually sums of squares of polynomials. Let us look at the simplest example where n = 2. The Hermite matrix of the (characteristic polynomial of the) symmetric 2 × 2 matrix

a b S = b c

34 is 2 a + c a + c a2 + 2b2 + c2 whose determinant is indeed a sum of squares (a − c)2 + 2b2. The paper by Borchardt gives the computation for n = 3. Much more on this kind of results can be found in [52]. One ﬁnds in [42, page 61] a theorem due to Sylvester and Hermite, also associating in a constructive way a symmetric matrix to a polynomial such that the number of real roots is given by the signature of the matrix. See Basu, Pollack and Roy [18, 4.3] for the applications of quadratic forms and matrices to the counting of polynomial roots in real algebraic geometry. The quotient (P (X)P 0(Y )−P (Y )P 0(X))/(X −Y ) is obviously a symmet- P i j ric polynomial of the form ai,jX Y and one associates the matrix ai,j to the polynomial P . This is called the Bezoutian symmetric matrix associated to P . It turns out that the signature of ai,j is also equal to the number of real roots of P . The connection between these two theorems is made clear for instance in [111]. For historical comments on the “once famous” Sturm’s theorem, we strongly recommend [19, 20, 21].

Hermite’s crater is next to Sylvester’s 1! 1et plus grand ! Note d’un des auteurs.

35 1.7 Witt groups of ﬁelds

The grave of Ernst Witt in the Nienstedtener Friedhof, Hamburg “Wo Kraft ist, wird auch die Zahl Meisterin: die hat mehr Kraft” (F. Nietzsche)

In the previous sections, we studied “classical” quadratic forms, defined on finite dimensional real vector spaces. By considering Sylvester’s application of forms to Sturm’s theorem we were naturally led to discuss symmetric matrices whose entries are polynomials in R[X] or more generally in R(X), the field or rational functions in one variable X, with real coefficients. It is therefore very natural to generalize as much as possible our considerations to symmetric matrices with entries in a general field K, or more generally in a commutative, or even non commutative, ring. In this subsection, we will begin with the most elementary case of a field K of characteristic different from 2, such as the special case of K = R(X). We will propose first an elementary down to earth approach and only mention more general results, referring to standard textbooks for a deeper description. The book of Lam [84] is a good general reference.

36 A symmetric form (V, φ) over K is a ﬁnite dimensional K-vector space V with a symmetric bilinear pairing φ : V × V → K. A symmetric form is essentially the same as V together with a quadratic function Q : V → K such that • Q(av) = a2Q(v) for a ∈ K, v ∈ V ,

• the function φ :(v1, v2) ∈ V ×V 7→ (Q(v1 +v2)−Q(v1)−Q(v2))/2 ∈ K is bilinear. Given (V, φ), define such a Q by Q(v) = φ(v, v)/2 ∈ K (v ∈ V ). A basis of V identifies a symmetric form (V, φ) with a symmetric matrix S with entries in K, such as we studied previously for K = R. There is a natural notion of isomorphism between symmetric forms and the goal of the theory is to obtain invariants of the isomorphism classes of symmetric forms over K. For K = R isomorphism is the same as the linear congruence we considered in section 1.2. The orthogonal subspace of a subspace U ⊆ V in a symmetric form (V, φ) is the subspace U ⊥ = {v ∈ V | φ(v, w) = 0 for all w ∈ U}. The radical of (V, φ) is V ⊥ ⊆ V . A symmetric form is nonsingular if its radical is trivial. For any (V, φ) there is defined a nonsingular symmetric form ⊥ (V/V , [φ]). A subspace U ⊆ V is called isotropic for (V, φ) if φ(v1, v2) = 0 for all v1, v2 ∈ U. A symmetric form is anisotropic if it contains no non trivial isotropic subspaces. A nonsingular symmetric form (V, φ) is called hyperbolic if it is the direct sum of two isotropic subspaces U and U 0, in which case φ restricts to an isomorphism

0 ∗ v1 ∈ U 7→ (v2 7→ φ(v1, v2)) ∈ U = HomK (U, K). The standard hyperbolic form is deﬁned for any ﬁnite-dimensional K-vector space U by H(U) = (U ⊕ U ∗, φ) with φ((v1, f1), (v2, f2)) = f1(v2) + f2(v1) ∈ K.

37 0 I For any basis on U and the dual basis on U ∗ the matrix of φ is n In 0 with n = dimK U. A nonsingular symmetric form over K is hyperbolic if and only if it is isomorphic to H(U) for some U. The direct sum of symmetric forms (V1, φ1), (V2, φ2) is the symmetric form (V1 ⊕ V2, φ1 ⊕ φ2), with

(φ1 ⊕ φ2)((v1, v2), (w1, w2)) = φ1(v1, w1) + φ2(v2, w2) ∈ K. Theorem 1.26 (Witt [148]). A symmetric form (V, φ) over any ﬁeld K (of characteristic 6= 2) is isomorphic to a sum (Va, φa)⊕(Vh, φh)⊕(V0, φ0) where:

• (Va, φa) is anisotropic,

• (Vh, φh) is hyperbolic,

• (V0, φ0) is trivial, i.e φ0 = 0. Moreover, this decomposition is unique, up to isomorphism. Since hyperbolic and trivial forms are classified by dimension, only the description of the anisotropic case is relevant. Definition 1.27. A diagonalization of a symmetric form (V, φ) over a field ∼ n K is an isomorphism (V, φ) = ⊕k=1(K, ak) with ak ∈ K. Recall that we already noticed the following fundamental fact 1.2. Proposition 1.28. Every symmetric form (V, φ) over a field K of characteristic 6= 2 can be diagonalized. For an anisotropic (V, φ) the symmetric matrix S with respect to any basis is regular, and the diagonalization is of the type ∼ n (V, φ) = ⊕k=1(K, µk(S)/µk−1(S)).

The dimension n = dimK V of a hyperbolic (V, φ) is even and there is a diagonalization of the type ∼ n/2 (V, φ) = ⊕k=1(K ⊕ K, 1 ⊕ −1). Proof. In view of Theorem 1.26 we can consider separately the anisotropic and hyperbolic cases. For an anisotropic (V, φ) the matrix S is regular, i.e. the principal minors µk(S) ∈ K (1 6 k 6 n) are non-zero. Proceeding as in the proof of

38 n Theorem 1.9 we have that (V, φ) is isomorphic to ⊕k=1(K, µk(S)/µk−1(S)). For the hyperbolic case it is enough to consider the 2-dimensional case 0 λ (V, φ) = (K ⊕ K, ) λ a for which there is deﬁned an isomorphism λ (a + 1)/2 :(V, φ) → (K ⊕ K, 1 ⊕ −1). λ (a − 1)/2

The set Sym(K) of isomorphism classes of nonsingular symmetric forms over K, equipped with ⊕, is an abelian monoid. Denote by GW (K) the corresponding Grothendieck group, i.e. the group of formal diﬀerences of symmetric forms. More precisely, it is the quotient of Sym(K) × Sym(K) by the equivalence relation (s1, s2) ∼ (t1, t2) if s1 ⊕ t2 is isomorphic to s2 ⊕ t1. Note that the natural map

(V, φ) ∈ Sym(K) 7→ ((V, φ), 0) ∈ GW (K) is injective so that there is no loss of information when we go from isomorphism classes to elements of GW (K). Deﬁnition 1.29. The Witt group of K is the quotient

W (K) = GW (K)/{subgroup generated by the hyperbolic forms}.

It follows from Witt’s theorem that the following conditions on two anisotropic nonsingular symmetric forms s1 = (V1, φ1), s2 = (V2, φ2) are equivalent:

• s1 is isomorphic to s2,

• s1 ⊕ t is isomorphic to s2 ⊕ t, for any nonsingular symmetric form t,

• s1 = s2 ∈ W (K). Therefore W (K) is indeed the key object in the understanding of nonsingular symmetric forms over K. Deﬁnition 1.30. A lagrangian L of a nonsingular symmetric form (V, φ) over F is a subspace L ⊆ V such that L = L⊥, with L⊥ = {x ∈ V | φ(x, y) = 0 ∈ F for all y ∈ L}.

39 Remark 1.31. The following conditions on a nonsingular symmetric form (V, φ) over K (of characteristic 6= 2) are equivalent.

1.( V, φ) is hyperbolic,

2.( V, φ) admits a lagrangian,

3.( V, φ) = 0 ∈ W (K).

In order to prove 3. =⇒ 2. one can appeal to Witt’s Theorem 1.26, or else note that if (V 0, φ0) is a nonsingular symmetric form over K with lagrangian L0 and L is a lagrangian of (V, φ) ⊕ (V 0, φ0) then V ∩ (L + L0) is a lagrangian of (V, φ). Before we go on, let us describe two trivial examples. We know the structure of nonsingular symmetric forms over R from The- n orem 1.1: every such form (V, φ) is isomorphic to ⊕k=1(R, ak) with ak = ±1. Since (R, 1) ⊕ (R, −1) is a hyperbolic form, we conclude that W (R) is isomorphic to Z, with the function

n τ = signature : (V, φ) = ⊕k=1(R, ak) ∈ W (R) 7→ τ(V, φ) ∈ Z an isomorphism. We know that every nonsingular symmetric form over the complex num- n bers C is isomorphic to ⊕k=1(C, 1), with (C, 1) ⊕ (C, 1) hyperbolic. The function

dim : (V, φ) ∈ W (C) 7→ dimC(V ) mod 2 ∈ Z/2Z is an isomorphism. In order to describe more interesting examples, we need to develop the basic features of Witt theory. Let K• = K\{0} the multiplicative group of units of K. Note that every element of K•/(K•)2 has order 2, with (K•)2 = {a2 | a ∈ K•} ⊆ K• the subgroup of squares. Note also that the group W (K) is denoted additively and K•/(K•)2 multiplicatively. First, we observe that there are two natural maps deﬁned on W (K).

Deﬁnition 1.32. 1. The dimension map

dim : (V, φ) ∈ W (K) 7→ dimK V mod 2 ∈ Z/2Z

40 is well-deﬁned for any K, since hyperbolic forms have even dimensions. 2. The discriminant map

disc : (V, φ) ∈ W (K) 7→ det(φ) ∈ K•/(K•)2 is well-deﬁned.

For instance, the 0 element of W (K) is the unique form on the 0-dimensional vector space and the associated 0 × 0 matrix has determinant 1! Note also that the discriminant is only a homomorphism when restricted to the kernel of the dimension map.

We can now give Witt’s description of W (K) by generators and relations, for any ﬁeld K of characteristic 6= 2.

Theorem 1.33 (Witt [148]). W (K) is isomorphic to Z[K•]/N with N ⊆ Z[K•] the subgroup generated by elements of the type [x] − [xy2] , [x] + [−x] , [x] + [y] − [x + y] − [xy(x + y)−1] for x, y ∈ K•, x + y 6= 0. The map

n n M X • (V, φ) = (K, ai) ∈ W (K) 7→ ai ∈ Z[K ]/N i=1 i=1 is an isomorphism.

Proof. We know that any anisotropic nonsingular symmetric form over K can be diagonalized, so that the map under consideration is well-defined and onto. The kernel will be identified with N by describing the relations in W (K) between these elements [x]. The first obvious relation is [x] = [xy2] for every y ∈ K•. This corresponds to the change of the basis element in a 1-dimensional vector space. Moreover, one has [−x] = −[x] as follows from the observation that [x] + [−x] is hyperbolic. Assume x + y 6= 0 and consider the matrices

x 0 1 −1 S = ; P = . 0 y −y x

41 One has x + y 0 P ∗SP = 0 xy/(x + y)) and it follows that

[x] + [y] = [x + y] + [xy/(x + y)] ∈ W (K).

Denote for a moment W¯ (K) the abstract abelian group generated by all [x]’s subject to these relations. One has a homomorphism W¯ (K) → W (K) which is onto and one has to show that it is 1 to 1. Indeed the group GL(n, K) is generated by matrices which act only non trivially on two vector basis. It follows that if a sum [x1] + ... + [xk] represents the zero element in W (K), i.e. if the corresponding diagonal form is isomorphic to a hyperbolic form, this isomorphism can be obtained by a chain of elementary isomorphisms which are described by the elementary operations described above. For more detail, see Lam [84]. Although the expression for W (K) given by Theorem 1.33 is not very useful for an arbitrary ﬁeld K, it is adequate for K in which it is known which elements of K• are squares: If K = R then x ∈ R• is a square if and only if x > 0, and the signed augmentation ∗ [x] ∈ W (R) = Z[R ]/N 7→ sign(x) ∈ Z is an isomorphism - evidently the signature! If K = C then every z ∈ C• is a square, so that [z] = [1] ∈ N. It now follows from ([z] + [i2z]) + ([z] − [−z]) = 2[z] = 0 ∈ N that the mod 2 augmentation • [z] ∈ W (C) = Z[C ]/N 7→ 1 ∈ Z/2Z is an isomorphism - evidently the dimension mod 2! See section 1.8 below for the examples K = R(X), Q, Fp. The case K = R(X) is particularly relevant to the Witt group interpretation of the theorems of Sturm and Sylvester.

1.8 The Witt groups of the function field R(X), ordered fields and Q. It is our goal to give an explicit description of the Witt group W (R(X)) of the field R(X) of rational functions P (X)/Q(X)(P (X),Q(X) ∈ R[X], Q(X) 6=

42 0). The inclusion R ⊂ R(X) induces a morphism W (R) = Z → W (R(X)) with image the Witt classes of quadratic forms with constant coefficients. We shall establish an isomorphism ∼ coker(Z → W (R(X))) = Z[R] ⊕ Z/2Z[H] with Z[R] the free abelian group generated by R, and similarly for Z/2Z[H] where H denotes the upper half plane {z ∈ C|=(z) > 0}. This will allow a modern presentation of Sturm and Sylvester’s theorems. Consider an invertible symmetric matrix S(X) over R(X). Its determinant is a non-zero rational function det(S(X)) ∈ R(X)•, with a finite number of zeroes and poles. Therefore, when x ∈ R is away from some finite set {x1, . . . , xk} of real numbers, the matrix S(x) is an invertible symmetric matrix in R, for which we can compute the signature τ(S(x)) ∈ Z. When x is a zero or a pole of the determinant of S(X), the value of τ(S(x)) is undefined. This suggests the following definition.

Definition 1.34. 1. A sign-function is a map R\{x1, . . . , xk} → Z whose domain of definition is the complement of a finite set {x1, . . . , xk} ⊂ R and which is constant on each connected component of the complement of this finite set. 2. Two sign-functions are equivalent if they agree outside of a finite set.

We denote the additive group of equivalence classes of sign-functions by Sign. A symmetric matrix S(X) in R(X) determines a sign-function

sign(S(X)) : x ∈ R\{poles of det(S(X))} 7→ τ(S(x)) ∈ Z.

∗ Note that if S2(X) = V (X)S1(X)V (X) , i.e. if S1(X) and S2(X) are linearly congruent under some invertible matrix V (X) with coefficients in R(X), one can conclude that the signatures of S1(x) and S2(x) coincide outside of the zeroes and poles of the determinant of V (X) so that two linearly congruent matrices do define equivalent sign-functions. Clearly the sign-function associated to a direct sum of two matrices is the sum of the sign-functions, and the sign-function of a hyperbolic matrix is obviously 0. Therefore, we have a well defined map:

sign : W (R(X)) → Sign.

43 More precisely, we will show that sign is injective when restricted to forms with discriminant 1. Before proving that, we will give a better description of the group Sign. Regard Z[R] as the group of functions R → Z which vanish outside of a ﬁnite set.

Lemma 1.35. There is an exact sequence

0 → Z → Sign → Z[R] → 0. Proof. Given a sign-function σ, one can deﬁne its “derivative”, i.e. the function ∂σ(a) = lim σ(a + ) − σ(a − ) →0 which is clearly an element of Z[R]. The kernel of ∂ consists of constant functions. One shows that ∂ is onto by “integration”.

Note that the injection Z → Sign splits in two natural ways, since any sign-function has well deﬁned limits at +∞ and −∞. Recall that the signature of any nonsingular symmetric form is always congruent modulo two to the dimension the space. Therefore, the image of the “derivative” ∂ ◦ sign : W (R(X)) → Z[R] is divisible by 2. We shall denote the “jump” (∂ ◦ sign)/2 by

τR : W (R(X)) → Z[R].

For n = 1 (i.e. for a 1 × 1 matrix), let F be a polynomial in R[X] and consider corresponding the element [F (X)] in W (R(X)). Of course, sign([F (X)]) is simply the usual sign of F (x), equal to ±1 outside of the set of roots of F . As for τR([F ]), it is non zero precisely at the values of x at which F (x) changes sign, from − to + or from + to −, with value ±1 accordingly. We will show that two invertible symmetric n × n matrices in R(X) are linearly congruent if and only if they have the same discriminant and the same sign-function. We now describe the discriminant of a nonsingular symmetric form over a ﬁeld K taking values in K•/(K•)2. Let us describe this abelian group when K = R(X).

44 Irreducible polynomials in R[X] have degree 1 or 2. Every a ∈ R determines an irreducible polynomial X −a ∈ R[X] with root a ∈ R. Every z ∈ H deﬁnes an irreducible real polynomial of degree 2:

2 2 Pz(X) = X − 2<(z)X + |z| . with complex roots z andz ¯. Any non-zero polynomial in R[X] can be written in a unique way as a product of irreducible polynomials, in the form

n m Y ni Y mj F (X) = λ (X − ai) Pzj (X) i=1 j=1 where λ 6= 0, ai ∈ R, zj ∈ H, and ni, mj > 0. A rational function F (X) can be written in the same way, if one agrees that ni and mj can be positive or negative. Such a function F is a square if and only if λ > 0 and all ni and mj are even numbers. It follows that we have a very simple description • • 2 ∼ R(X) /(R(X) ) = Z/2Z ⊕ Z/2Z[R] ⊕ Z/2Z[H] and the discriminant map has three components

• • 2 disc = (disc1, discR, discH): W (R(X)) → R(X) /(R(X) ) .

We can now give a precise description of the Witt group W (R(X)). As before, if P (X)/Q(X) ∈ R(X)• we denote by [P (X)/Q(X)] the element of W (R(X)) deﬁned by the 1 × 1 matrix with the single element P (X)/Q(X). Theorem 1.36. The following map is an isomorphism

p : Z[1] ⊕ Z[R] ⊕ Z/2Z[H] → W (R(X)) X X (n, na, mz) 7→ n[1] + na[X − a] + mz[Pz(X)]. a∈R z∈H It is easy to show that p is injective. Indeed,

• sign([X −a]) is equal to −1 for x < a and 1 for x > a so that τR([X −a]) is the function equal to 1 for x = a and 0 for x 6= a.

• sign([1]) and sign([Pz(X)]) are constant functions equal to 1 so that

τR([1]) = τR(Pz(X)) = 0.

45 • disc(p(n, na, mz)) = (n, na, mz) ∈ Z/2Z ⊕ Z/2Z[R] ⊕ Z/2Z[H].

Any element of the kernel of p has therefore na = 0. It then follows that n = 0 and ﬁnally mz = 0. It remains to show that p is onto. In other words, we have to show that W (R(X)) is generated by [1], [X − a], [Pz(X)]. We already know that elements of the form [F (X)] generate W (R(X)). Note that:

• [P (X)/Q(X)] = [P (X)Q(X)] so that the [F (X)]’s with F polynomial in R[X] generate W (R(X)). • Modulo squares, it is enough to consider the F (X) of the form

n m Y Y F (X) = ± (X − ai) Pzj (X) i=1 j=1

with distinct ai, zj, so that ni = mj = 1.

We can now show that 1, [X − a] and [Pz(X)] generate W (K). The proof will be by induction. Let us ﬁrst consider products of two factors. Let P (X) and Q(X) be two irreducible polynomials, each of the form (X −a) or Pz(X). It is easy to check that one can ﬁnd two non zero real numbers λ, µ such that λP (X) + µQ(X) is a square in R[X]. Therefore we have, in W (K(X)): λµP Q [λP ] + [µQ] = [λP + µQ] + . λP + µQ so that ±[P ] ± [Q] = [1] ± [PQ]. So, the product of two irreducible polynomials [PQ] can be expressed as a combination of those polynomials [P ], [Q], and [1]. By induction, we can therefore express any [F (X)] as a combination of the elements 1, [X −a] and [Pz(X)]. This ends the proof of Theorem 1.36 giving a full description of W (R(X). Let us reformulate it in the following way:

46 Corollary 1.37. The Witt group of R(X) ﬁts into a split exact sequence

0 → Z → W (R(X)) → Z[R] ⊕ Z/2 Z[H] → 0 where the ﬁrst arrow is induced by the embedding R ⊂ R[X] and the second is τR ⊕ discH.

Note that we already showed that the injection Z → W (R(X)) splits in two ways, using limits of sign at ±∞. The following is just a reformulation of what we proved so far.

Corollary 1.38. Let S(X) be a symmetric n × n matrix in R(X) with non vanishing determinant, so that (R(X)n,S(X)) is a nonsingular symmetric form over R(X) with a Witt class [S(X)] ∈ W (R(X)). For any a ∈ R the coeﬃcient τR([S(X])(a) is half the jump in the signature lim(τ(S(a + )) − τ(S(a − )))/2 −→ of the nonsingular symmetric forms (Rn,S(x)) over R. If a ∈ R is not a zero or a pole of the determinant of S(X), there is no jump.

Now that we have a complete understanding of the Witt group over R(X), we can restate Sylvester’s reformulation of Sturm’s theorem, as an equality between elements in a Witt group. For any degree n regular polynomial P (X) ∈ R[X] with Sturm functions (P0(X),P1(X),...,Pn(X)), (Q1(X),Q2(X),...,Qn(X)), denote by [T ri(Q)] the Witt class of the tridiagonal nonsingular symmetric form (R(X)n, Tri(Q(X))) over R(X). The group identity

n M [T ri(Q)] = (R(X),Pk(X)/Pk−1(X)) k=1 follows from the proof of Theorem 1.9, or else one checks directly that the following matrix (from Barge and Lannes [17, p.2]) is an isomorphism of

47 symmetric forms over R(X)

 1 0 0 ... 0   −1 −1 0 ... 0     1 1 1 ... 0    :  ......   . . . . .  (−1)n+1 (−1)n+1 (−1)n+1 ... (−1)n+1 n L n (R(X),Pk(X)/Pk−1(X)) → (R(X) , Tri(Q)). k=1 Let us evaluate

n X τR([T ri(Q)]) = τR(Pk(X)Pk−1(X)). k=1

We have to add signs ± for each root of one of the polynomials Pk(X)Pk−1(X), depending on whether Pk(X)Pk−1(X) is increasing or decreasing in the neighborhood of this root. Note that

• Any root of Pk for 1 6 k 6 n−1 appears twice in P0P1,P1P2,...,Pn−1Pn.

• Pk−1(x) and Pk+1(x) have opposite signs at any root of Pk since Pk−1(X)+ Pk+1(X) = Pk(X)Qk(X). Therefore the two signs associated to the two occurrences of some root of Pk cancel (1 6 k 6 n − 1).

0 • The product P0(X)P1(X) = P (X)P (X) is increasing in the neighbor- 0 02 00 hood of any root of P0 (since the derivative of PP is P + PP ).

• Pn is a non zero constant so does not vanish! It follows that ( 0 if P (x) 6= 0 τ ([T ri(Q)])(x) = R +1 if P (x) = 0 so that τR([T ri(Q)]) is indeed counting the roots of P (with a + sign). We therefore recover the theorem of Sturm and Sylvester 1.22, 1.24.

Corollary 1.39. For any a < b ∈ R the composite

τ restriction augmentation W (R(X)) R / Z[R] / Z[[a, b]] / Z

48 sends (R(X)n, Tri(Q(X))) to {no. of real roots of P (X) in [a, b]} = (τ(T (b)) − τ(T (a)))/2 = var(a) − var(b) ∈ {0, 1, . . . , n}.

Remark 1.40. Our description of the sign-function sign on W (R(X)) can be expressed in a slightly diﬀerent language. An ordering O of K is a decomposition as a disjoint union

K = K+ t {0} t K− such that

• x ∈ K− if and only if −x ∈ K+,

• The sum and product to two elements of K+ belongs to K+. The O-sign of an element a ∈ K is  +1 if x ∈ K  + signO(a) = 0 if x = 0  −1 if x ∈ K− and satisfies the usual product rules for signs. The proof of Sylvester’s Law of Inertia 1.3, with no modification, enables us to define the O-signature of a symmetric (V, φ) using any of the diagonal- ∼ n ization (V, φ) = ⊕k=1(K, ak) given by Proposition 1.28. The number is inde- 2 • 2 pendent of the diagonalization, since a square b ∈ K has signO(b ) = +1 for every ordering O. The O-signature of a nonsingular symmetric form (V, φ) over K is

n X τO(V, φ) = signO(ak) ∈ Z k=1 ∼ n for any diagonalization (V, φ) = ⊕k=1(K, ak). In our new language, any ordering O on K yields a signature homomorphism (also congruent modulo 2 to the dimension)

n n X τO : ⊕k=1(K, ak) ∈ W (K) 7→ signO(ak) ∈ Z. k=1

49 In the special case of K = R(X), we leave to the reader the description of orderings and to relate the corresponding O-signatures to the map sign : W (R(X)) → Sign that we constructed above. Pfister [107] proved the great theorem: for any field K of characteristic 6= 2 the intersection of all kernels of the homomorphisms τO with O running over all the orderings of K is torsion. In other words, for every (V, φ) ∈ W (K) of infinite order, there is some ordering O such that τO(V, φ) 6= 0.

The description that we gave of W (R(X)) “by hand” can be greatly generalized and conceptualized. We refer to [84, Chapter 6] for a detailed exposition and we limit ourselves to a quick sketch since we shall not use this generalization. Let K be any field (of characteristic different from 2) and let us compute W (K(X)). Any polynomial in K[X] splits as the product of a constant c and irreducible monic polynomials. Let π ∈ K[X] be such an irreducible monic polynomial. The quotient K[X]/π is the residue field at π. As usual in commutative algebra, one can construct the localization of K[X] at π, inverting all elements which are not divisible by π. This produces a ring Kπ[X] whose field of fractions is K(X). For instance, when K = R and π = X − a, the ring Kπ[X] consists of rational functions P (X)/Q(X) which are finite in a neighborhood of a, i.e. such that Q(a) 6= 0. So, the passage from K[X] to Kπ[X] is indeed a “localization around a”. K(X) is equipped with a discrete valuation vπ with values in Z ∪ {∞} n such that vπ(π P (X)/Q(X)) = n if P,Q are coprime to π. The subring consisting of elements with non negative valuation is by definition the local ring Kπ[X]. The norm exp(−vπ) defines a topology on K(X) which is not complete. Denote its completion by Kπ(X). For instance, when K = R and π = X − a, the residue field K[X]/π is isomorphic to R. As for the field Kπ(X), it consists of Laurent series at a, i.e. formal sums +∞ X k αk(X − a) k=v with αk ∈ R and αv 6= 0. The valuation of such an element is v ∈ Z. Note that series whose valuation is 0 are precisely the invertible elements of the local ring Kπ[X], i.e. of the ring of elements with non negative valuation. In

50 P+∞ k general, the field Kπ(X) consists of series k=v αkπ where the coefficients αk are now in the residue field K[X]/π. We define a “residue map”

resπ : W (K(X)) → W (K[X]/π). The construction is easy as a composition of two natural maps. The ﬁrst, W (K(X)) → W (Kπ(X), is simply induced by the embedding of K(X) in its completion Kπ(X). The second goes from W (Kπ(X)) to the Witt group of its residue ﬁeld W (K[X]/π). Denote by U the group of unit elements of Kπ[X] (with valuation 0). Modulo squares, a non zero element q of Kπ(X) can be written in the form u or πu for some unit

+∞ X k u = αkπ k=0 with α0 ∈ K[X]/π is different from 0. One sets resπ([u]) = 0 and resπ([πu]) = [α0] ∈ W (K[X]/π). In the case K = R, and π = X − a the residue map resa : W (K(R)) → −1 W (R) is such that resa(α−1(X − a) ) is +1 or −1 according to the sign of α−1 and this is coherent with the classical concept of residue in complex analysis. In effect, Milnor ([97, Chapter IV]) established the following exact sequence for any field K of characteristic 6= 2, generalizing our computation for K = R M 0 → W (K) → W (K(X)) → W (K[X]/π) → 0. πirr The proof follows the same line as the special case that we examined in detail. The case of the rationals Q is another example where this general tech- nique applies. Prime ideals of Z correspond to prime numbers p. The residue fields are Fp. The completion of the field of fractions Q of Z for the p-adic valuation is the field Qp of p-adic numbers. The completion of Q with the archimedean valuation is of course R, with Witt group Z. The residue map goes from W (Q) to W (Fp) and the corresponding exact sequence is now

0 → Z → W (Q) → Z/2 ⊕ ⊕p6=2W (Fp) → 0.

51 See Milnor and Husemoller [97, Chapter III] for many computations of the symmetric Witt group W (K). In particular, the Witt group of a ﬁnite ﬁeld Fq with q odd is Z/4Z if q is 3 mod 4 and isomorphic to the group ring • • 2 (Z/2Z)[Fq/(Fq) ] if q = 1 mod 4.

52 2 Topology

2.1 Even-dimensional manifolds We shall only consider compact oriented smooth manifolds, possibly with a non empty boundary, and in the first instance work with homology and cohomology with coefficients in a commutative ring R. Recall that for any commutative ring R the cup product equips the R- coefficient cohomology of any space X M H?(X; R) = Hk(X; R) k>0 with a canonical structure of a graded associative and commutative ring. Commutativity should be understood in the graded sense. For a closed (i.e. without boundary) oriented m-dimensional manifold M there is a Poincaréduality, which can be expressed as an isomorphism k between H (M; R) and Hm−k(M; R). Using this duality, the ring structure on cohomology then transforms to an intersection product between homology

53 classes: φM : Hk(M; R) × H`(M; R) → Hk+`−m(M; R). The intersection in homology is also well deﬁned for a manifold M with k ∼ boundary ∂M, using the Poincar´e-Lefschetz duality H (M; R) = Hm−k(M, ∂M; R). For k +` = m composition with the augmentation map H0(M; R) → R gives a bilinear pairing

φM : Hk(M; R) × H`(M; R) → H0(M; R) → R such that k` φM (v, u) = (−1) φM (u, v) ∈ R for u ∈ Hk(M; R) and v ∈ H`(M; R). We shall be particularly concerned with the cases R = Z, R. In this case, the homology abelian groups Hk(M; Z) are f.g., and the homology vector spaces Hk(M; R) are finite dimensional. Remark 2.1. A smooth oriented submanifold L` ⊂ M represents a Z-coefficient k ` m homology class [L] ∈ H`(M, Z). For submanifolds K ,L ⊂ M the algebraic intersection φM ([K], [L]) ∈ Hk+`−m(M; Z) has a geometric description: by a small deformation, one can assume that the two submanifolds are transverse and one considers the Z-coefficient homology class of the oriented intersection. If Kk,L` ⊂ M m are transverse then K ∩ L ⊆ M is a (k + ` − m)-dimensional submanifold (understood to be ∅ if k + ` < m) and

φM ([K], [L]) = [K ∩ L] ∈ Hk+`−m(M; Z). For k + ` = m the intersection K ∩ L is a ﬁnite set of points, and the augmentation H0(M; Z) → Z (an isomorphism for connected M) sends φM ([K], [L]) ∈ H0(M; Z) to the number of points, with a sign given by the orientations. Therefore, restricting to the case of the middle dimension, there is a canonical ±-symmetric form associated to an even dimensional manifold.

Deﬁnition 2.2. For any 2`-dimensional manifold with boundary (M, ∂M) there is deﬁned a (−1)`-symmetric pairing

φM : H`(M; R) × H`(M; R) → R.

54 ` If H`(M; R) is a f.g. free R-module this is an example of a (−1) - symmetric form over R, an obvious generalization of the symmetric forms over a field F we were considering in section 1. Definition 2.3. Let = ±1. 1. A -symmetric form over R (V, φ) is a f.g. free R-module V with a bilinear pairing φ : V × V → R such that φ(v, w) = φ(w, v). 2. A form (V, φ) is nonsingular if the R-module morphism φ : V → V ∗ = HomR(V,R) given by φ(v)(w) = φ(v, w) is an isomorphism. 3. An isomorphism f :(V, φ) → (V 0, φ0) of -symmetric forms over R is an R-module isomorphism f : V → V 0 such that φ0(f(v), f(w)) = φ(v, w). (In the Appendix 7 we shall even be considering forms over a ring with involution). Let F`(M; Z) denote the quotient of H`(M; Z) by its torsion. This is a f.g. free Z-module which is also equipped with the (−1)`-symmetric intersection ` form φM . The intersection matrix of (M, ∂M) is the integral (−1) -symmetric n × n matrix SM = φM (bi, bj) ∈ Z with respect to some basis {b1, b2, . . . , bn} ⊂ F`(M; Z). The intersection matrix is invertible for closed M or if ∂M = S2`−1, in which case it has determinant ±1 and the form (F`(M; Z), φM ) over Z is nonsingular. In 1923, Hermann Weyl [145] defined the signature of an oriented 4k- dimensional manifold with boundary (M, ∂M) as the signature of the intersection symmetric n × n matrix SM

τ(M) = τ(SM ) ∈ Z. See Eckmann [43] for the backstory of Weyl’s paper. This is Weyl’s signature:

Thom [135] proved that any homology class in H`(M; Z) has an integral multiple which is represented by an `-dimensional submanifold P ` ⊂ M. Thus for any basis {b1, b2, . . . , bn} ⊂ F`(M; Z) there exist non zero integers k1, k2, . . . , kn such that kibi = [Pi] ⊂ F`(M; Z) for some `-dimensional

55 ` submanifolds Pi . The intersection number φM (bi, bj) is therefore equal to φM ([Pi], [Pj])/kikj. As mentioned earlier, the intersection φM ([Pi], [Pj])/kikj can be interpreted as a number of signed intersections of Pi and Pj in general position. The intersection matrix SM determines the intersection form over R since R ⊗Z F`(M; Z) = H`(M; R). Very frequently, we shall forget the integral ` structure on H`(M; Z) and only consider the (−1) -symmetric form φM on the vector space H`(M; R). Its radical is

⊥ ∗ H`(M; R) = ker(φM : H`(M; R) → H`(M; R) ) = im(H`(∂M; R)) ⊆ H`(M; R).

Example 2.4. The 2k-dimensional complex projective space CP2k is a closed 2k 4k-dimensional manifold with the homology H2k(CP ; Z) = Z generated by the fundamental class of some CPk linearly embedded in CP2k with self- intersection 1. The intersection 1 × 1 matrix is the symmetric 1 × 1 matrix

SCP2k = (1), so that the signature is 2k τ(CP ) = 1 ∈ Z. Note that if one reverses the orientation of M, the signature is changed to its opposite, τ(−M) = −τ(M). Note that when the dimension of the manifold has the form 2` = 4k + 2, the intersection form is skew symmetric and therefore there is no notion of signature. In order to understand the intersection form on the boundary of an odd- dimensional manifold we recall some standard algebraic notions: Deﬁnition 2.5. 1. Let (V, φ) be an -symmetric form over R. Given a submodule L ⊆ V let L⊥ = {x ∈ V | φ(x, y) = 0 ∈ R for all y ∈ L} ⊆ V . The submodule is isotropic if L ⊆ L⊥. An isotropic submodule is a sublagrangian if L and L⊥ are f.g. free direct summands of V .A lagrangian L is a sublagrangian such that L = L⊥, in which case (V, φ) is nonsingular. 2. The -symmetric Witt group W(R) is the group of equivalence classes of nonsingular -symmetric forms over R, with (V, φ) ∼ (V 0, φ0) if and only if (V, φ) ⊕ (H, θ) is isomorphic to (V 0, φ0) ⊕ (H0, θ0) for some forms (H, θ), (H0, θ0) with lagrangians.

56 Of course, if R = F is a ﬁeld and = 1 this is just the Witt group W (F ) already considered in section 1.

Proposition 2.6. 1. For a (2` + 1)-dimensional manifold with boundary (N, ∂N) the submodule

L = ker(H`(∂N; R) → H`(N; R)) ⊂ H`(∂N; R)

` is isotropic with respect to the (−1) -symmetric pairing φ∂N : H`(∂N; R) × H`(∂N; R) → R. 2. If R is a ﬁeld then L is a lagrangian of the nonsingular (−1)`- symmetric form (H`(∂N; R), φ∂N ) over R. In particular, for R = R, ` = 2k this gives τ(∂N) = 0 ∈ Z. Proof. Consider the commutative diagram with exact rows

H`+1(N, ∂N; R) / H`(∂N; R) / H`(N; R)

H`(N; R) / H`(∂N; R) / H`+1(N, ∂N; R)

∗ ∗ ∗ H`(N; R) / H`(∂N; R) / H`+1(N, ∂N; R)

Example 2.7. Consider M 2` = S` × S` = ∂N with N 2`+1 = D`+1 × S`. The intersection (−1)`-symmetric form of M is hyperbolic

0 1 (H (M; R), φ ) = H ` (R) = (R ⊕ R, ) ` M (−1) (−1)` 0 with H`(M; R) = R ⊕ R, generated by the two factors intersecting in one point and each having self intersection 0. The lagrangian determined by N is

L = ker(H`(M; R) → H`(N; R)) = R ⊕ 0 ⊂ H`(M; R) = R ⊕ R.

By deﬁnition, a space X is k-connected for some k > 1 if it is connected and satisﬁes one of the equivalent conditions:

57 1. πj(X) = 0 for 1 6 j 6 k,

2. π1(X) = {1} and Hj(X; Z) = 0 for 1 6 j 6 k. We shall say that an m-dimensional manifold with boundary (M, ∂M) is k-connected if M is k-connected and ∂M is connected. For an (`−1)-connected 2`-dimensional manifold with boundary (M, ∂M) we have that the Z-module H`(M; Z) is f.g. free, with the natural Z-module morphism deﬁned by evaluation

` ∗ f ∈ H (M; Z) 7→ (u 7→ f(u)) ∈ H`(M; Z) an isomorphism. The intersection pairing thus deﬁnes a (−1)`-symmetric form (H`(M; Z), φM ) over Z. Furthermore, Hj(∂M; Z) = 0 for 1 6 j 6 `−2, and in view of the exact sequence

φM ∗ 0 / H`(∂M; Z) / H`(M; Z) / H`(M; Z) / H`−1(∂M; Z) / 0 we have that :

1.( H`(M; Z), φM ) is nondegenerate (i.e. det(φM ) 6= 0 ∈ Z) if and only if H`(∂M; Z) = 0, if and only if ∂M is a Q-coeﬃcient homology (2` − 1)- sphere,

2.( H`(M; Z), φM ) is nonsingular if and only if H`(∂M; Z) = H`−1(∂M; Z) = 0, if and only if ∂M is a Z-coeﬃcient homology (2` − 1)-sphere.

2.2 Cobordism Deﬁnition 2.8. An (m+1)-dimensional cobordism (N; M,M 0) is an (m+1)- dimensional manifold N with the boundary decomposed as ∂N = M t −M 0 for closed (m − 1)-dimensional manifolds M, M 0, where −M 0 = M 0 with the opposite orientation.

Signature is cobordism invariant:

Theorem 2.9 (Thom [135]). If (N; M,M 0) is a (4k + 1)-dimensional cobordism then 0 τ(M) − τ(M ) = τ(∂N) = 0 ∈ Z.

58 Proof. As in Proposition 2.6 2., for any ﬁeld R the subspace

L = ker(H2k(∂N; R) → H2k(N; R)) ⊂ H2k(∂N; R) is a lagrangian of the intersection symmetric form

0 (H2k(∂N; R), φ∂N ) = (H2k(M; R), φM ) ⊕ (H2k(M ; R), −φM 0 ).

For R = R we thus have that ∂N = M t −M 0 has signature τ(∂N) = 0. The set of cobordism classes of closed n-dimensional manifolds is an abelian group Ωn, with addition by disjoint union, and the cobordism class of the empty manifold ∅ as the zero element. Thom [135] initiated the computation of Ωn, starting with the signature

τ :Ω4k → W (R) = Z; M 7→ τ(M) = (H2k(M; R), φM )

It turns out that τ is an isomorphism for k = 1 and a surjection for k > 2. Each Ωn is finitely generated. Recall that every vector bundle over any space M canonically defines characteristic classes, which are cohomology classes in M (Milnor and Stash- eff [99]). With integral coefficients, this leads to the Pontrjagin classes pk which belong to H4k(M; Z). This applies in particular to the tangent bundle of a smooth closed manifold M and produces the Pontrjagin classes of the manifold. If 4k = 4k1 + 4k2 + ··· + 4ki, the class pk1 pk2 . . . pki is in H4k(M; Z) = Z (for connected M) and is therefore an integer: one speaks of the Pontrjagin numbers of the manifold M. They are invariant under cobordism. For more on cobordism theory, see Milnor and Stasheff [99]. The signature is clearly a homotopy invariant. The Pontrjagin classes are diffeomorphism invariants but not homeomorphism invariants, let alone homotopy invariants. It was therefore a surprise when in 1953, Hirzebruch [62], [64] showed that the signature of a closed 4k-dimensional manifold M is expressed in terms of characteristic numbers. More precisely, there is a formula: Z τ(M) = L(M) = hLk(p1, p2, . . . , pk), [M]i ∈ Z ⊂ R M where Lk is a certain polynomial in the Pontrjagin classes. For instance,

1 • L1 = 3 p1

59 1 2 • L2 = 45 (7p2 − p1)

1 3 • L3 = 945 (62p3 − 13p1p2 + 2p1)

1 2 2 4 • L4 = 14175 (381p4 − 71p1p3 − 19p2 + 22p1p2 − 3p1)

For a Mathematica code generating the Lk’s, see McTague [92] In 1956 Milnor [93] used the failure of the Hirzebruch signature theorem for manifolds with boundary to detect that certain 7-dimensional manifolds homeomorphic to S7 had exotic (i.e. non-standard) differentiable structures; the subsequent work of Kervaire and Milnor [79] used the signatures of manifolds with boundary as essential tools in the classification of all high-dimensional exotic spheres. 2 In 1965 Novikov [103] used signatures of non-compact manifolds to prove that the rational Pontrjagin classes 4k pk(M) ∈ H (M; Q) are homeomorphism invariants. A different approach to the Hirzebruch signature theorem is due to Atiyah and Singer [12]. In the 1960’s they proved the “Atiyah-Singer Index Theo- rem” expressing the analytic index of an elliptic operator on a closed manifold in terms of characteristic classes. The signature is the index of the signature operator: the index theorem in this case recovers the Hirzebruch signature theorem. The proof of the index theorem is a piece of cake (designed by John Roe and baked by Ida Thompson for Michael Atiyah’s 75th birthday in 2004):

2See the October 2015 issue of the Bulletin of the American Mathematical Society for a collection of materials relating to the history of manifolds in general and exotic spheres in particular.

60 2.3 Odd-dimensional manifolds Just as forms are the basic algebraic invariants of even-dimensional manifolds, so formations are the basic algebraic invariants of odd-dimensional manifolds.

Deﬁnition 2.10. (Ranicki [117, p.68]) Let = ±1. 1. An -symmetric formation (H, θ; F,G) over R is a nonsingular - symmetric form (H, θ) over R together with a lagrangian F and a sublagrangian G. 2. An isomorphism of formations

f :(H, θ; F,G) → (H0, θ0; F 0,G0) is an isomorphism of forms f :(H, θ) → (H0, θ0) such that f(F ) = F 0, f(G) = G0.A stable isomorphism of formations

[f]:(H, θ; F,G) → (H0, θ0; F 0,G0) is an isomorphism of the type

f :(H, θ; F,G) ⊕ (K, λ; I,J) → (H0, θ0; F 0,G0) ⊕ (K0, λ0; I0,J 0) with K = I ⊕ J, K0 = I0 ⊕ J 0. 3. The boundary of the formation is the nonsingular -symmetric form

∂(H, θ; F,G) = (G⊥/G, [θ]).

4. The formation is nonsingular if ∂(H, θ; F,G) = 0, i.e. if G is a lagrangian. 5. The boundary of an -symmetric form (V, φ) over R is the nonsingular (−)-symmetric formation over R

∂(V, φ) = (H−(V ); V, {(u, φ(u)) | u ∈ V }).

Proposition 2.11. 1. A nonsingular -symmetric form (V, φ) (over any R) is such that (V, φ) = 0 ∈ W(R) if and only if (V, φ) is (isomorphic to) the boundary ∂(H, θ; F,G) of an -symmetric formation (H, θ; F,G) over R. 2. If (H, θ; F,G) is an -symmetric formation over a ﬁeld R then

L = ((F + G) ∩ G⊥)/G ⊆ G⊥/G

61 is a lagrangian of the nonsingular -symmetric form ∂(H, θ; F,G) = (G⊥/G, [θ]). 3. An automorphism α :(H, θ) → (H, θ) of a nonsingular -symmetric form (H, θ) with a lagrangian F ⊂ H determines a nonsingular -symmetric formation (H, θ; F, α(F )). 4. If 1/2 ∈ R for any nonsingular -symmetric formation (H, θ; F,G) there exists an automorphism α :(H, θ) → (H, θ) such that α(F ) = G.

Proof. 1. Immediate from the extension of Proposition 7.3 1. from a ﬁeld to a ring. 2. Immediate from 1. and Remark 1.31 2. 3. Trivial. 4. Immediate from the extension of Proposition 7.3 2.

We denote by Σn the orientable surface of genus n.

Σ1 Σ3

Σ2

Note that the intersection form of Σn is the standard symplectic form 0 In n n ∗ n ∗ n ∗ φΣn = : H1(Σn; R) = R ⊕ R → H1(Σn; R) = (R ) ⊕ (R ) −In 0

The mapping class group Γn = π0(Aut(Σn)) is the group of orientation preserving diﬀeomorphisms α :Σn → Σn, modulo isotopy. The symplectic group of a ring R n Sp(2n, R) = Aut(H−(R )) (n > 1) consists of the invertible 2n × 2n matrices A = (aij)(aij ∈ R) preserving the standard symplectic form on R2n:

0 I 0 I A∗ n A = n . −In 0 −In 0

62 Looking at the action of α ∈ Γn on homology, one gets a canonical group homomorphism: cn :Γn → Sp(2n, Z) which is an isomorphism for n = 1 and a surjection for n > 2. Any α ∈ Γn determines a closed 3-dimensional manifold with a Heegaard decomposition

3 1 2 1 2 N = #nS × D ∪α #nS × D and hence a nonsingular skew-symmetric formation over R which only depends on c(α) ∈ Sp(2n, Z): n n (H, θ; L1,L2) = (H1(Σn; R), φΣn ; R ⊕ 0, γnc(α)(R ⊕ 0)). Every closed connected 3-dimensional manifold N has a Heegaard decomposition; two decompositions determine a stable isomorphism of formations. Remark 2.12. The first application of the symplectic group in topology is a b due to Poincaré[109]: the construction for any α = ∈ Sp(2, ) = c d Z SL(2, Z) of the 3-manifold 3 1 2 1 2 N = S × D ∪α S × D , which is in fact the lens space L(c, a). See Chapter 6.3 below for a somewhat more detailed account of the lens spaces, and the connection with number theory. Definition 2.13. A generalized Heegaard decomposition of a closed (` − 1)- connected (2` + 1)-dimensional manifold N 2`+1 is an expression as

N = N1 ∪M N2 for a closed (` − 1)-connected codimension 1 submanifold M 2` ⊂ N and (` − 1)-connected codimension 0 submanifolds N1,N2 ⊂ M with

M = N1 ∩ N2 = ∂N1 = ∂N2 ⊂ N

63 2`+1 Proposition 2.14. A generalized Heegaard decomposition N = N1∪M N2 ` determines a nonsingular (−1) -symmetric formation (H`(M; R), φM ; L1,L2) with the lagrangians

L1 = ker(H`(M; R) → H`(N1; R)), L2 = ker(H`(M; R) → H`(N2; R)) ⊂ H`(M; R) such that

L1 ∩ L2 = H`+1(N; R),H`(M; R)/(L1 + L2) = H`(N; R) Proof. Immediate from the Mayer-Vietoris exact sequence

0 → H`+1(N; R) → H`(M; R) → H`(N1; R) ⊕ H`(N2; R) → H`(N; R) → 0.

Every closed (` − 1)-connected (2` + 1)-dimensional manifold N has a generalized Heegaard decomposition; the formations associated to different decompositions are stably isomorphic. Remark 2.15. 1. For even-dimensional manifolds the intersection (−1)`-symmetric intersection form defines a function {diffeomorphism classes of (` − 1)-connected 2`-dimensional manifolds with boundary} → {isomorphism classes of (−1)`-symmetric forms over Z}; (M, ∂M) 7→ (H`(M; Z), φM ) with closed manifolds sent to nonsingular forms. See Wall [141] for the classification of (` − 1)-connected 2`-dimensional manifolds for ` > 3. 2. For odd-dimensional manifolds generalized Heegaard decompositions give a corresponding function {diffeomorphism classes of (` − 1)-connected (2` + 1)-dimensional manifolds with boundary} → {stable isomorphism classes of (−1)`-symmetric formations over Z}; (N, ∂N) 7→ (H, θ; L1,L2) with L1 ∩ L2 = H`+1(N; Z),K/(L1 + L2) = H`(N; Z), ⊥ (L2 /L2, [θ]) = (H`(∂N), φ∂N ), and closed manifolds sent to nonsingular formations. See Wall [142] for the classification of (` − 1)-connected (2` + 1)-dimensional manifolds for ` > 3.

64 2.4 The union of manifolds with boundary; Novikov additivity and Wall nonadditivity of the signature In Theorem 2.18 below we state the Novikov additivity theorem for the signature of a union of 4k-dimensional manifolds glued along components. We then go on to state in Theorem 2.27 the Wall nonadditivity for the signature of a union of 4k-dimensional relative cobordisms. Deﬁnition 2.16. 1. An m-dimensional manifold with boundary (M, ∂M) is a union of m- dimensional manifolds with boundary (M1, ∂M1), (M2, ∂M2) along a codimension 1 submanifold with boundary (N, ∂N) ⊂ (M, ∂M) if

(M, ∂M) = (M1, ∂M1) ∪(N,∂N) (M2, ∂M2), that is if (M, ∂M) = (M1 ∪N M2,N1 ∪∂N N2) with

N1 = cl.(∂M1\N),N2 = cl.(∂M2\N), ∂M1 = N ∪∂N N1, ∂M2 = N ∪∂N N2, ∂N = ∂N1 = ∂N2

2. A union is special if

∂N = ∅, ∂M1 = N, ∂M2 = N t N2,N1 = ∅,N2 = ∂M. 3. The trinity of the union in 1. is the m-dimensional manifold with boundary

T = T (∂N; N,N1,N2) = {neighbourhood of the stratiﬁed set N ∪ N1 ∪ N2 ⊂ M} 2 = ∂N × D ∪∂N×(I0tI1tI2) (N × I0 t N1 × I1 t N2 × I2) ⊂ M

65 2 1 with I0,I1,I2 ⊂ ∂D = S disjoint closed arcs.

The closure of the complement

0 0 0 0 (cl.(M\T ), ∂(cl.(M\T ))) = (M1, ∂M1) t (M2, ∂M2)

0 0 0 0 is the disjoint union of two copies (M1, ∂M1), (M2, ∂M2) of (M1, ∂M1), 0 0 (M2, ∂M2), with ∂T = ∂M t (∂M1 t ∂M2). This gives (M, ∂M) an expression as a special union

0 0 0 0 (M, ∂M) = (M t M , ∂M t ∂M ) ∪ 0 0 (T, ∂T ). 1 2 1 2 (∂M1t∂M2,∅) Remark 2.17. The trinity terminology is due to Borodzik, Nemethi and Ran- icki [26], which studies relative cobordisms of manifolds with boundary using the method of algebraic surgery theory. We have the Novikov additivity theorem for the signature: Theorem 2.18 ((Novikov [104] 1967 for ∂N = ∅, Wall [143] 1969 in general). The signature of a 4k-dimensional manifold with boundary which is a union

(M, ∂M) = (M1, ∂M1) ∪(N,∂N) (M2, ∂M2) is the sum of the signatures of M1,M2 and the trinity T

τ(M) = τ(M1) + τ(M2) + τ(T ) ∈ Z. In particular, if ∂N = ∅ (e.g. if the union is special) then τ(T ) = 0. Proof. Consider ﬁrst the case of an expression of a 4k-dimensional manifold with boundary as a special union

(M, ∂M) = (M1, ∂M1) ∪(N,∂N) (M2, ∂M2).

66 The subforms (V1, φ1), (V2, φ2) of the nonsingular symmetric form over R

∗ (V, φ) = (im(φM : H2k(M; R) → H2k(M; R) ), ΦM ) (ΦM (φM (u), φM (y)) = φM (u, v)) deﬁned by

∗ (V1, φ1) = (im(H2k(M1; R) → H2k(M; R) ), φ|), ∗ (V2, φ2) = (im(H2k(M2; R) → H2k(M; R) ), φ|) ⊆ (V, φ)

⊥ are such that V1 = V2. The diagonal

∆ : u ∈ (V1, 0) 7→ (u, u) ∈ (V1, −φ1) ⊕ (V, φ) is the inclusion of a sublagrangian with

⊥ (∆ /∆, [−φ1 ⊕ φ]) = (V2, φ2).

By Proposition 7.3 (in the Appendix 7)

−τ(V1, φ1) + τ(V, φ) = τ(V2, φ2) ∈ Z so that

τ(M) = τ(V, φ) = τ(V1, φ1) + τ(V2, φ2) = τ(M1) + τ(M2) ∈ Z For the general case note that the union

0 0 0 0 (M, ∂M) = (M t M , ∂M t ∂M ) ∪ 0 0 (T, ∂T ) 1 2 1 2 (∂M1t∂M2,∅) is special, so that

0 0 τ(M) = τ(V, φ) = τ(M1 t M2) + τ(T ) = τ(M1) + τ(M2) + τ(T ) ∈ Z.

Finally, if ∂N = ∅ then T = (N t N1 t N2) × [0, 1] and τ(T ) = 0 ∈ Z.

In 1969 Wall [143] computed the signature of the trinity T = T (∂N; N,N1,N2) of a 4k-dimensional union in terms of the ‘triformation’ consisting of the symplectic form over R

(H, θ) = (H4k−2(∂N; R), φN )

67 and the three lagrangians

L = ker(H4k−2(∂N; R) → H4k−2(N; R)), L1 = ker(H4k−2(∂N; R) → H4k−2(N1; R)), L2 = ker(H4k−2(∂N; R) → H4k−2(N2; R)). by constructing a symmetric form W (H, θ; L, L1,L2) over R such that

τ(T ) = τ(W (H, θ; L, L1,L2)) ∈ Z. This nonadditivity of the signature invariant has subsequently turned out to be a manifestation of the Maslov index, as we shall recount further below. Deﬁnition 2.19 (Ranicki, [117]). 1.An -symmetric form (V, φ) over R is a union of subforms (V1, φ1), (V2, φ2) ⊆ (V, φ) (V, φ) = (V1, φ1) ∪ (V2, φ2) ⊥ ⊥ if φ(V1,V2) = {0}, or equivalently V1 ⊆ V2 , V2 ⊆ V1 . ⊥ 2. The union is special if V1 = V2. Example 2.20. Let (M, ∂M) be a 2`-dimensional manifold with boundary ` with (−1) -symmetric intersection form (V, φ) = (H`(M; R), φM ) over R. An expression as a union

(M, ∂M) = (M1, ∂M1) ∪(N,∂N) (M2, ∂M2) determines submodules

V1 = im(H`(M1; R) → H`(M; R)),V2 = im(H`(M2; R) → H`(M; R)) ⊆ V such that φ(V1,V2) = {0}. If V1 and V2 are f.g. free direct summands of V (e.g. if R is a field) then (V, φ) is the union of the subforms (V1, φ1), (V2, φ2) in the sense of Definition 2.19 1., with φ1 = φ|V1 , φ2 = φ|V2 . Furthermore if (M, ∂M) is a special geometric union in the sense of Definition 2.16 2. and (V, φ) is a special algebraic union in the sense of Definition 2.19 2.

Proposition 2.21. For a special union -symmetric form (V, φ) = (V1, φ1)∪ (V2, φ2) there is deﬁned an isomorphism of -symmetric forms ∼ (V3, φ3) ⊕ (V1, φ1) = (V2, −φ2) ⊕ (V, φ) for some nonsingular -symmetric form (V3, φ3) with a lagrangian.

68 Proof. As in the proof of Theorem 2.18 apply Theorem 7.3 to the diagonal

∆ : (u, u) ∈ (V2, 0) 7→ (u, u) ∈ (V2, −φ2) ⊕ (V, φ) which is the inclusion of a sublagrangian with

⊥ (∆ /∆, [−φ2 ⊕ φ]) = (V1, φ1).

Example 2.22. For a special union symmetric form (V, φ) = (V1, φ1)∪(V2, φ2) over R τ(V, φ) = τ(V1, φ1) + τ(V2, φ2) ∈ Z. Deﬁnition 2.23. 1. A (−)-symmetric triformation over R (H, θ; L1,L2,L3) is a nonsingular (−)-symmetric form (H, θ) over R together with three lagrangians L1,L2,L3 ⊂ H. 2. The boundary of the triformation is the nonsingular (−)-symmetric formation

∂(H, θ; L1,L2,L3) = (H, θ; L1,L2) ⊕ (H, θ; L2,L3) ⊕ (H, θ; L3,L1).

3.The Wall form ([143]) W (H, θ; L1,L2,L3) = (W, ψ) is the -symmetric pairing ψ : W × W → R;(u, v) 7→ ψ(u, v) = ψ(v, u) with

W = ker((j1 j2 j3): L1 ⊕ L2 ⊕ L3 → H), ji = inclusion : (Li, 0) → (H, θ), . ψ : W × W → R;(u, v) = ((u1, u2, u3), (v1, v2, v3)) 7→ θ(j1(u1), j2(v2))

4. Let (V, φ) be an -symmetric form over R which is a union

(V, φ) = (V1, φ1) ∪ (V2, φ2),

⊥ ⊥ such that V1 ⊆ V2 , V2 ⊆ V1 are direct summands. Let

i1 :(V1, φ1) → (V, φ), i2 :(V2, φ2) → (V, φ)

69 be the inclusions. The canonical (−)-symmetric triformation (H, θ; L1,L2,L3) over R is given by i∗φ i∗ ker( 1 1 : V ⊕ V ∗ → V ∗ ⊕ V ∗) 0 i∗ 1 2 0 1 (H, θ) = 2 , , − 0 i1 i2 ∗ im( : V1 ⊕ V2 → V ⊕ V ) −φi1 0 ⊥ ∗ ∗ 1 V1 ker(i1φ : V → V1 ) : L1 = = → K, 0 V2 im(i2 : V2 → V ) ⊥ ∗ ∗ 0 V2 ker(i2φ : V → V2 ) : L2 = = → K, 1 V1 im(i1 : V1 → V ) ∗ 0 i1 ∗ ∗ ∗ : L3 = ker( ∗ : V → V1 ⊕ V2 ) → K 1 i2

Example 2.24. 1. Deﬁnition 2.23 3. associates to any skew-symmetric formation (H, θ; L1,L2,L3) over R a symmetric form W (H, θ; L1,L2,L3) over R and hence a signature

τ(H, θ; L1,L2,L3) = τ(W (H, θ; L1,L2,L3)) ∈ Z.

2. For any α1, α2 ∈ Sp(2n, R) we have a skew-symmetric triformation (H−(L); L, α1(L), α2(L)) and hence a symmetric form W (H−(L); L, α1(L), α(L2)) over R. For R ⊆ R the signature

τ(W (H−(L); L, α1(L), α2(L))) ∈ W (R) = Z is a cocycle on Sp(2n, R) : these are the ‘Meyer cocycle’ for R = Z and the ‘Maslov cocycle’ for R = R – see section 4 below. Proposition 2.25. 1. Let (H, θ; L1,L2,L3) be an (−)-symmetric triformation over R. If W = ker(L1 ⊕ L2 ⊕ L3 → K) is a f.g. free R-module (e.g. if R is a ﬁeld, or if K = L1 + L2 + L3) then the Wall form (W, ψ) is an -symmetric form over R, with the boundary stably isomorphic to the boundary of the triformation ∼ ∂(W, ψ) = ∂(H, θ; L1,L2,L3).

2. A union -symmetric form (V, φ) = (V1, φ1) ∪ (V2, φ2) over R is such that there is deﬁned an isomorphism of -symmetric forms over R ∼ (V, φ) = ((V1, φ1) ⊕ (V2, φ2)) ∪ W (H, θ, L1,L2,L3)

70 with (H, θ; L1,L2,L3) the canonical (−)-symmetric triformation over R given by Deﬁnition 2.23. 3. For = 1, R = R

τ(V, φ) = τ(V1, φ1) + τ(V2, φ2) + τ(W (H, θ, L1,L2,L3)) ∈ Z. Proof. An abstract version of Wall [143]. Example 2.26. 1. Let 2` (M , ∂M) = (M1, ∂M1) ∪(N,∂N) (M2, ∂M2) be an (` − 1)-connected 2`-dimensional manifold with boundary which is a union of (` − 1)-connected manifolds with boundary (M1, ∂M1), (M2, ∂M2) along an (` − 2)-connected manifold with boundary (N, ∂N). The (−1)`- symmetric intersection form (V, φ) = (H`(M; R), φM ) is an algebraic union

(V, φ) = (V1, φ1) ∪ (V2, φ2) = (H`(M1; R), φM1 ) ∪ (H`(M2; R), φM2 ) such that the canonical (−1)`−1-symmetric triformation is determined by ∂N = ∂N1 = ∂N2

(H, θ; L, L1,L2) = (H`−1(∂N; R), φ∂N ; ker(H`−1(∂N; R) → H`−1(N; R)), ker(H`−1(∂N; R) → H`−1(N1; R)), ker(H`−1(∂N; R) → H`−1(N2; R))) The Wall form of the triformation is the (−1)`-symmetric intersection form of the trinity T = T (∂N; N,N1,N2)

W (H, θ; L, L1,L2) = (H`(T ; R), φT ). 2. For ` = 2k, R = R

τ(M) = τ(M1) + τ(M2) + τ(T ) = τ(M1) + τ(M2) + τ(W (H, θ; L, L1,L2)) ∈ Z with τ(T ) = τ(W (H, θ; L, L1,L2)) ∈ Z the nonadditivity invariant of Wall [143]. Finally, here is the Wall non-additivity theorem in full generality: Theorem 2.27 (Wall [143]). The signature of a 4k-dimensional manifold with boundary which is a union

(M, ∂M) = (M1, ∂M1) ∪(N,∂N) (M2, ∂M2) is given by τ(M) = τ(M1) + τ(M2) + τ(T ) ∈ Z with T = T (∂N; N,N1,N2) the trinity of the union.

71 2.5 Plumbing: from quadratic forms to manifolds

The algebraic plumbing of symmetric matrices over R (Deﬁnition 1.8) extends readily to -symmetric forms over Z:

Deﬁnition 2.28. The plumbing of an -symmetric form (V, φ) over Z with respect to v ∈ V ∗ and a 1-dimensional -symmetric form (Z, w) is the - symmetric form over Z φ v∗ (V 0, φ0) = (V ⊕ , ). Z v w

Proposition 2.29. If (V 0, φ0) is an -symmetric form over Z which is obtained from an -symmetric form (V, φ) over Z with det(φ) 6= 0 ∈ Z by plumbing with v ∈ V ∗, (Z, w) then up to isomorphism

0 0 −1 ∗ Q ⊗Z (V , φ ) = Q ⊗Z (V, φ) ⊕ (Z, w − vφ v ).

Proof. Immediate from the isomorphism of -quadratic forms over Q

−1 ∗ 1 −φ v =∼ : ⊗ (V 0, φ0) / ⊗ (V, φ) ⊕ ( , w − vφ−1v∗) 0 1 Q Z Q Z Q

We shall now show that geometric plumbing determines algebraic plumbing of the intersection form over Z, and that in certain cases algebraic plumbing is realized by the geometric plumbing procedure of Milnor. Denote by SO(n) the subgroup of the orthogonal group consisting of matrices of determinant 1. Recall that any continuous map w : S`−1 → SO(n) can be used to construct a Rn-vector bundle over the sphere S`. The sphere S` can be decomposed as two hemispheres, homeomorphic to the ball D` intersecting along the equator, homeomorphic to S`−1. One then starts with two trivial bundles D` × Rn and one glues them on their boundaries, iden- tifying the point (u, v) ∈ S`−1 × Rn in the boundary of the ﬁrst copy to (u, w(u)(v)) ∈ S`−1 × Rn in the boundary of the second copy. The map w is called a clutching map. Any oriented vector bundle over a sphere is isomorphic to one obtained by this construction so that the set of isomorphism classes of vector bundles over S` is identiﬁed with the homotopy group

72 π`(BSO(n)) = π`−1(SO(n)). The Bott periodicity theorem gave the computation of the stable groups

π (SO) = lim π (SO(n)), `−1 −→ `−1 n leading to the computation of the unstable groups (Kervaire [77]). The Euler number (deﬁned as the self-intersection of the zero section of the corresponding `-plane bundle over S`) deﬁnes a homomorphism

χ : w ∈ π`−1(SO(`)) 7→ χ(w) ∈ Z. Note that χ(w) = 0 if ` is odd. One can also think of χ(w) as the degree of the map S`−1 → S`−1 sending u ∈ S`−1 to the ﬁrst column of w(u), and also the Hopf invariant of a map J(w): S2`−1 → S`. For ` = 2k and k 6= 1, 2, 4 the Euler number is even since the Hopf invariant of any map S4k−1 → S2k is even in these dimensions (Adams [2]). We now describe a general plumbing construction of 2`-dimensional manifolds with boundary, using surgery. Start with the following input:

1.a2 `-dimensional manifold with boundary (M, ∂M),

2. an embedding v :(D` × D`, S`−1 × D`) ⊆ (M, ∂M),

3. a clutching map w : S`−1 → SO(`) producing a vector bundle E(w) over S`. The plumbed 2`-dimensional manifold with boundary is the following out- put:

0 0 ` ` `−1 ` ` `−1 (M , ∂M ) = (M ∪f(w) D × D , cl.(∂M\S × D ) ∪ D × S ), f(w): S`−1 × D` → S`−1 × D`;(u, v) 7→ (u, w(u)(v))

73 M = M0 [ M1

` ` ` ` M0 = cl.(MnM1) M1 = v(D ¢ D ) M2 = D ¢ D

0 M = M0 [ M1 [f (w) M2

` ` M1 [f (w) M2 = D -bundle of w over S

Let us observe the algebraic eﬀect of geometric plumbing in the highly- connected case of interest: Proposition 2.30. Let (M, ∂M) be an (` − 1)-connected 2`-dimensional manifold with boundary, and let (M 0, ∂M 0) be the plumbed manifold with boundary obtained using

` ` `−1 ` `−1 v :(D × D , S × D ) ⊂ (M, ∂M), w : S → SO(`). 1. (M 0, ∂M 0) is (`−1)-connected with (−1)`-symmetric intersection form over Z given by algebraic plumbing (Deﬁnition 2.28)

` ? 0 φM (−1) v (H (M ; ), φ 0 ) = (H (M; ) ⊕ , ) ` Z M ` Z Z v χ(w) where ` ` ? v = v[D × D ] ∈ H`(M, ∂M; Z) = H`(M; Z) and χ(w) = (−1)`χ(w) ∈ Z is the Euler number of w.

741 0 2. If (H`(M; Z), φM ) is nondegenerate then (H`(M ; Z), φM 0 ) is nonde- ` −1 ∗ generate if and only if w − (−1) vφM v 6= 0 ∈ Q, in which case

0 ` −1 ∗ Q ⊗Z (H`(M ; Z), φM 0 ) = Q ⊗Z (H`(M; Z), φM ) ⊕ (Q, χ(w) − (−1) vφM v ). For even ` it follows that the signatures of M, M 0 are related by

0 −1 ? τ(M ) = τ(M) + sign(χ(w) − vφM v ) ∈ Z. Remark 2.31. In the situation of Proposition 2.30 2. with ` even we have a union as in Deﬁnition 2.16

0 0 ` ` ` ` (M , ∂M ) = (M, ∂M) ∪(S`−1×D`,S`−1×S`−1) (D × D , ∂(D × D )).

The skew-symmetric triformation over Q given by Example 2.26 is 1 vφ−1v∗ χ(w) (H ( ); , M , ) − Q 0 Q 1 Q 1 Q

−1 ? with the Wall form (W, ψ) = (Q, χ(w) − vφM v ) of signature

−1 ? τ(W, ψ) = sign(χ(w) − vφM v ) ∈ Z. Deﬁnition 2.32. A graph manifold is an (` − 1)-connected 2`-dimensional manifold with boundary constructed from D`×D` by the geometric plumbing of n `-plane bundles over S`, using a graph with vertices j = 1, 2, . . . , n and ` weights χj ∈ π`−1(SO(`)). The weights are `-plane bundles χj over S .

Theorem 2.33 (Milnor 1959 [94], Hirzebruch 1961 [66]). Let ` > 2 and let ` S = (sij ∈ Z) be a (−1) -symmetric n × n matrix, such that for ` = 2k and k 6= 1, 2, 4 the diagonal entries sjj ∈ Z are even. Then S is realized by a graph 2`-dimensional manifold with boundary (M, ∂M) such that

n (H`(M; Z), φM ) = (Z ,S).

If the graph is a tree then for ` > 2 M is (` − 1)-connected, and for ` > 3 M and ∂M are both (` − 1)-connected. The plumbing construction and graph manifolds were motivated by Hirze- bruch’s 1950’s and 1960’s work on the resolution of singularities (see section 6.3 below for one particular class of examples realizing tridiagonal symmetric matrices) and by the exotic spheres Σn of Milnor [93], [94].

75 The celebrated 8 × 8 symmetric matrix over Z 2 1 0 0 0 0 0 0 1 2 1 0 0 0 0 0   0 1 2 1 0 0 0 0   0 0 1 2 1 0 0 0 E8 =   0 0 0 1 2 1 0 1   0 0 0 0 1 2 1 0   0 0 0 0 0 1 2 0 0 0 0 0 1 0 0 2 is invertible, positive deﬁnite, has even diagonal entries, and signature

τ(E8) = 8 ∈ Z. See the website Ranicki [113] for source material on the role of the number 8 in general and the form E8 in particular, in topology, algebra and mathematical physics.

The 4-dimensional graph manifold (M, ∂M) with intersection matrix E8 has 3 3 3 boundary the Poincar´ehomology 3-sphere ∂M = Σ with H∗(Σ ) = H∗(S ), 3 π1(Σ ) 6= {1}. If Σ2`−1 = ∂M for a parallelizable (` − 1)-connected (M 2`, ∂M) then for ` > 3 the diﬀerentiable structure on ∂M is detected by the signature

76 defect of (M, ∂M) if ` is even, and by the Kervaire invariant if ` is odd. In particular, for k > 2 the (2k − 1)-connected 4k-dimensional graph manifold (M, ∂M) with intersection matrix E8 has boundary the Milnor exotic sphere 4k−1 ∂M = Σ generating the finite cyclic group bP4k of the exotic spheres which bound framed manifolds (Milnor [93], Kervaire and Milnor [79]). 2` 2`−1 Brieskorn [29] proved that all such pairs (M , Σ ) for ` > 3 are fibres of the Milnor fibration [95]

2`+1 1 u ∈ S \V 7→ f(u)/|f(u)| ∈ S determined by an isolated singular point of a complex hypersurface V = f −1(0) ⊂ C`+1, with f a polynomial function

` `+1 X aj f :(u0, u1, . . . , u`) ∈ C 7→ uj ∈ C. j=0

77 3 Knots, links, braids and signatures

3.1 Knots, links, Seifert surfaces and complements

1 If c > 1 is an integer, we shall denote by Sc the disjoint union of c copies of 1 3 the circle. A (classical) c-component link is a smooth embedding L : Sc ,→ S 1 of Sc in the 3-sphere. A knot is a link with c = 1. Two c-component links L0,L1 are considered equivalent if they are isotopic, i.e. if there is a smooth family (Lt)t∈[0,1] connecting them. Equivalently, they are isotopic if and only if there is an orientation preserving diffeomorphism of the 3-sphere sending L0 on L1 (as follows from Cerf’s theorem that the group of orientation preserving diffeomorphisms of the 3-sphere is connected). A link is trivial if it is isotopic to a link whose image lies in R2 ⊂ R3 ⊂ S3. There are many excellent books presenting the present state of knot and link theory, including from an historical perspective. We recommend in particular the books of Crowell and Fox [38], Sossinsky [128], Kauffman [74] and Lickorish [88]. In this section, we would like to concentrate on a very specific part of this theory relating knots and links to quadratic forms and their signatures. The first step is actually related to many other aspects of knot theory: the Seifert surfaces originally defined for knots, but later extended to links. In 1930 Frankl and Pontrjagin [47] published the result that every knot k : S1 ,→ S3 is the boundary k(S1) = ∂F of an embedded (oriented) surface F,→ S3. This result was reproved in 1935 by Seifert [123], acknowledging the priority of [47]. The advantage of [123] is that it provided a practical algorithm for constructing a surface from a knot k projection: this is called a Seifert surface for k. A knot is isotopic to the trivial knot if and only if there exists a Seifert surface which is a disc. The algorithm applies to any 1 3 3 1 link L : Sc ,→ R , and produces a surface F,→ R such that ∂F = L(Sc ). There exists a linear map P : R3 → R2 (many such in fact) such that the 1 2 image of the composite PL : Sc → R is a collection of oriented curves with a finite number ` of transverse double points labelled as over/underpasses. This is a plane projection of L. Modify this collection to delete all double points to obtain n Seifert circles as shown on the following figure, extracted from [128, page 21].

78 Each of these curves bounds a disc in the plane. One can push these discs away from the plane, using the third dimension, in such a way that they are disjoint. Corresponding to each double point of the projected knot, one glues a twisted band between the corresponding pushed discs. The result is a Seifert surface with n 0-handles and ` 1-handles

a 2 a 1 1 3 F = D ∪ D × D ⊂ R n ` such that 1 3 ∂F = L(Sc ) ⊂ R . Seifert used this construction to associate a bilinear form to a knot (which also applies to a link). Consider a Seifert surface F for k : S1 ⊂ S3. If u and v are two homology classes of H1(F ; Z), one can represent them by two closed curves in F . Pushing u in the positive normal direction of F , one gets two disjoint curves u+ and v in the 3-sphere and the linking number lk(u+, v) ∈ Z is therefore well deﬁned. This is a non symmetric bilinear form called the Seifert form

Σ: H1(F ; Z) × H1(F ; Z) → Z. Note that lk(u+, v) − lk(v+, u) is the (skew symmetric) intersection of the ∗ two curves u, v on F , so that (H1(F ; Z), Σ − Σ ) is a symplectic form over Z.

79 See the paper of Collins [37] for the relationship between the genus of F , Σ, c, ` and n. As an example, the following picture, extracted from [74, page 200] shows a connected Seifert surface for the trefoil knot, a basis a, b for its homology and the Seifert matrix associated to this basis.

The main difficulty with this construction is of course that the Seifert surface is not unique, so that this form is not canonically attached to the knot. Further below we shall describe the effect on the Seifert matrix Σ of choosing a different Seifert surface F . It is not hard to prove that one can go from one connected Seifert surface F1 to another one F2 by very simple elementary operations. The first is simply an isotopy of the ambient sphere. The second is a 1-surgery: delete two disjoint discs in the interior of Seifert surface F and connect their boundaries by a tube disjoint from F . The third is the opposite move: choose an embedded tube in F , cut it open, and fill the two circles with discs. One can look at the effect of these operations on the Seifert matrices. If the Seifert matrix associated to a Seifert surface F is an n × n matrix Σ, after a 1-surgery it is an (n + 2) × (n + 2) matrix of the form

Σ 0 0 0 Σ = 0 0 1 . α 0 0

80 It was therefore natural to look for invariants of Seifert matrices under this kind of operations which generate the S-equivalence relation, which was introduced by Murasugi [102]. Equivalently, one can study invariants of the homology of the canonical inﬁnite cyclic cover of the link exterior. 1 3 The exterior of a link L : Sc ,→ S is the 3-dimensional manifold X 1 1 with boundary ∂X = L(Sc ) × S obtained by deleting from the 3-sphere a tubular neighborhood of the link. The inclusion of the link exterior in the 3 1 link complement X,→ S \L(Sc ) is a homotopy equivalence. The linking number with L deﬁnes a canonical surjection π1(X) → Z which represents (1, 1,..., 1) ∈ H1(X) (which is a direct sum of c copies of Z). The kernel of this epimorphism is the fundamental group

π1(X) = ker(π1(X) → Z) of an inﬁnite cyclic covering X of X. One way to deﬁne this covering is to choose a map p : X → S1 represent- ing this epimorphism, and to identify X with the pullback

2πit 1 X = {(x, t) ∈ X × R | p(x) = e ∈ S } of the universal cover of the circle. The homeomorphism

z :(x, t) ∈ X 7→ (x, t + 1) ∈ X is a generating covering translation, and the lift of the projection p

p :(x, t) ∈ X 7→ t ∈ R is Z-equivariant. Choose p : X → S1 to be a smooth map whose restriction to the boundary ∂X = k(S1) × S1 is the second projection. The inverse image of a regular value ∗ ∈ S1 is a Seifert surface F = p−1(∗) ,→ S3 for k with ∂F = k(S1) × {∗} ,→ X. The embedded surface F,→ X has a tubular neighbourhood F × [0, 1] ,→ X. The covering X is of course trivial over X \ F . The closure −1 of one component ofp ¯ (X \ F ) ,→ X is a compact manifold XF whose boundary consists of two copies of F , one being the image of the other by z. Equivalently, XF = closure(X\(F × [0, 1])) ,→ X. Note that XF is a fundamental domain for the covering X → X.

81 X

1 1 2 z Fz XF XF zFF zXF z F

z 1k(S1) ¢ I k(S1) ¢ I zk(S1) ¢ I z 1k(S1) k(S1) zk(S1) z 2k(S1)

A curve in F and a curve in X \ F are disjoint curves in S3, and have a well deﬁned linking number in Z. This deﬁnes a Z-module isomorphism

=∼ ∗ H1(XF ; Z) / H1(F ; Z) ; u 7→ (y 7→ lk(u, v)).

The inclusions i+ : F,→ XF and i− : zF ,→ XF induce dual Z-module morphisms

∼ ∗ (i+)∗ = Σ : H1(F ; Z) → H1(XF ; Z) = H1(F ; Z) , ∗ ∼ ∗ (i−)∗ = Σ : H1(F ; Z) → H1(XF ; Z) = H1(F ; Z) and there is deﬁned a short exact sequence of Z[z, z−1]-modules

∗ −1 Σ−zΣ ∗ −1 0 / H1(F ; Z)[z, z ] / H1(F ; Z) [z, z ] / H1(X; Z) / 0 .

This construction is due to Hirsch and Neuwirth [61]. It follows from p∗ : ∼ 1 −1 H∗(X) = H∗(Sc ) that in the knot case c = 1 the Z[z, z ]-module morphism 1 − z : H1(X) → H1(X) is an isomorphism. The Alexander polynomial of k

? ∆k(z) = det(zΣ − Σ ) ∈ Z[z] is such that ∆k(z)H1(X; Z) = {0}. The Alexander polynomial is an isotopy invariant of the link L, which was introduced in [4] in a much more algebraic (and obscure) way. The expression of ∆k(z) in terms of Σ is due to Seifert [123]. The modern version of ∆k(z) is the Alexander-Conway polynomial, the determinant of uΣ − u−1Σ∗ in Z[u, u−1] (with u = z2), which is related to the Jones polynomial.

1 82 3.2 Signatures of knots : a tale from the sixties

Seifert [123] used his matrix Σ to compute the homology groups H∗(Xa) (in a fact, only the Betti numbers) of the ﬁnite cyclic covers Xa = X/{z } (a > 1) of the knot exterior X, which are isotopy invariants. In 1962, Trotter [138] published a remarkable paper providing a detailed analysis of the homology H∗(X) of X itself. In particular, he observed that for any Seifert matrix Σ the signature of the symmetric matrix Σ + Σ? over Z is also invariant ([138, Proposition (v)1]). This is the signature of the knot. In general, the matrix Σ + Σ∗ is only invertible over Q. In 1965, Murasugi [102] used this signature to get concrete topological consequences. For instance, he showed that an alternate knot has a positive signature and therefore cannot be isotopic to its mirror image since one easily checks that mirror images have opposite signatures. More importantly in our context, he showed that the signature can be used to study cobordism. This concept had recently been introduced in paper by Fox and Milnor [46] (which actually only appeared in 1966). A knot is slice if it bounds an embedded disc in the 4-ball. Two knots 1 3 k0, k1 are cobordant if one can embed a cylinder S × [0, 1] in S × [0, 1] whose boundary is k0 × {0} and k1 × {1}. If a knot is slice, it is easy to show that the homology of a Seifert surface contains a subspace of half dimension on which the Seifert form identically vanishes. It follows indeed that the signature vanishes. Two independent papers by Tristram [137] and Levine [86] appeared in 1969 and extended this idea, both deﬁning the ω-signature of a Seifert matrix Σ of a knot k : S1 ,→ S3 for ω ∈ S1 ⊂ C to be the signature of the hermitian form (1 − ω)Σ + (1 − ω)Σ?

? τω(Σ) = τ((1 − ω)Σ + (1 − ω)Σ ) ∈ Z, establishing cobordism invariance for appropriate ω. The determinant of the hermitian form is such that

? det((1 − ω)Σ + (1 − ω)Σ ) = −(1 − ω)∆k(ω) ∈ C so that for ω 6= 1 ∈ S1 the hermitian form is nonsingular if and only if ∆k(ω) 6= 0 ∈ C. In [137] only the following ω were considered

ωp = exp((p − 1)πi/p) for an odd prime p, and ω2 = exp(πi) = −1

83 and it was proved that ∆k(ωp) 6= 0 and that the ωp-signatures τωp (k) ∈ Z are cobordism invariants. In [86] it was proved that the ω-signature function

1 τ : ω ∈ S 7→ τω(Σ) ∈ Z takes constant values between the roots ω ∈ S1 of the Alexander polynomial ∆k(z), and that these values are cobordism invariants. See section 3.3 below for a discussion of the jumps in the ω-signature function. The connected sum of knots defines an operation which provides the space of cobordism of knots with the structure of an abelian group C1. Tris- ∞ tram [137] uses these ω signatures to find some epimorphism from C1 to Z . The independent paper by Levine [86] contains similar results, expressed in a more algebraic terminology, but also showing that the ω-signatures determine the cobordism class of a high-dimensional knot k : S2i−1 ,→ S2i+1 modulo torsion - see Ranicki [118] for an account of the high-dimensional knot cobordism groups using the L-theory of Z[z, z−1] with the involution z¯ = z−1. Viro [139] and Kauffman and Taylor [75] identified τ(k) with the signature of the 4-dimensional manifold N which is a double cover of D4 branched along k. See more details in Kauffman [74] and Lickorish [88].

3.3 Two great papers by Milnor and the modern approach The 1968 paper by J. Milnor [98] is a pure gem and contains the “right definition” of the Murasugi-Tristram-Levine signatures. There is no more any need for the non canonical choice of a Seifert surface and one gets quadratic forms canonically attached to a knot. As a motivation, let us introduce first the concept of fibred knot. A knot k : S1 ,→ S3 is fibred its complement has the structure of an open book. In other words, there should exist a locally trivial fibration p : S3\k(S1) → S1. In a tubular neighborhood of k, identified with S1 × D2, the map p is defined outside k and should be of the form p(x, z) = Arg(z) ∈ S1. For an arbitrary ∗ ∈ S1 the fibre p−1(∗) is such that Σ = k(S1) ∪ p−1(∗) is a Seifert surface for k. The fibres of p look indeed like the pages of a book whose binding is k. Many interesting knots are fibred. For instance, if P (u, v) is a polynomial in two complex variables (u, v) such that (0, 0) is an isolated singularity of the curve P (u, v) = 0, one can consider the knot k which is the intersection

84 3 2 2 2 of P (u, v) = 0 with a small sphere S = |u| + |v| = and the map p = P (u, v)/|P (u, v)| ∈ S1 provides such a ﬁbration. See the book by Milnor [95] on singularities of hypersurfaces. However, many knots are not ﬁbred. As before, let

3 1 2 1 1 (X, ∂X) = (closure(S \k(S ) × D ), k(S ) × S ) and let (X, ∂X) be the canonical inﬁnite cyclic cover. In the case where k is ﬁbred, (X, ∂X) = (F, ∂F ) × R with the generating covering translation z : X → X given by

z :(x, t) ∈ F × R 7→ (A(x), t + 1) ∈ F × R for some monodromy diffeomorphism A : F → F , which is the identity in a neighborhood of the boundary. Such a picture does not hold in the general non fibred case, but it does hold at the homology level, at least if one works with real coefficients. Milnor shows the following, for any knot k:

• H1(X, ∂X, R) has a ﬁnite even dimension, 2 ∼ • H (X, ∂X, R) = R, • The evaluation of the cup product

1 1 2 ∼ θ : H (X, ∂X, R) × H (X, ∂X, R) → H (X, ∂X, R) = R

defines a symplectic vector space (H1(F ; R), θ) over R. Let us call a linear automorphism A fibred if A − I is invertible, i.e. if 1 is not an eigenvalue. The monodromy acts on the symplectic vector space (H1(X, ∂X; R), θ) by a fibred linear automorphism, still denoted by A, such that A∗θA = θ. In this way, we can attach canonically a fibred symplectic automorphism A of a symplectic vector space to any knot k. This is the fundamental invariant of a knot. For any fibred automorphism A of a symplectic vector space (H, θ) over R there is defined an abstract Seifert form (= asymmetric bilinear form) over R Σ = θ(I − A)−1 : H → H∗

85 such that A = Σ−1Σ∗, with

Σ − Σ∗ = θ : H → H∗.

1 This allows the ω-signature τω(A) ∈ Z to be deﬁned for any ω ∈ S to be the signature of the hermitian form

∗ ∗ (1 − ω)Σ + (1 − ω)Σ : C ⊗R H → C ⊗R H as in section 3.2. For any knot k it is possible to choose a Seifert surface F ∗ such that the Seifert form Σ : H1(F ; R) → H1(F ; R) is an isomorphism, in which case the exact sequence

∗ −1 Σ−zΣ ∗ −1 0 / H1(F ; R)[z, z ] / H1(F ; R) [z, z ] / H1(X; R) / 0 gives an isomorphism of symplectic forms over R ∗ ∼ 1 (H1(F ; R), Σ − Σ ) = (H (X; R), θ) and the ﬁbred automorphism of the symplectic form (H1(X; R), θ) over R is given by −1 ∗ 1 1 A = Σ Σ :(H (X; R), θ) → (H (X; R), θ) and τω(k) = τω(A). In the second step, one should extract invariants from symplectic automorphisms. As we know, any symplectic vector space is isomorphic with R2n equipped with the bilinear form:

Ω((x1, . . . , x2n), (y1, . . . , y2n)) = x1yn+1 + . . . xny2n − xn+1y1 − · · · − x2nyn.

An 2n × 2n matrix A is said to be symplectic if it preserves the bilinear form Ω. The symplectic group

n Sp(2n, R) = AutRH−(R ) consists of the symplectic 2n × 2n matrices - we shall describe it in more detail later on. Therefore, any knot deﬁnes a symplectic matrix, unique up to conjugacy. Conjugacy invariants of matrices, real or complex, are very well known and are described by the classical Jordan normal forms. In the generic case, when eigenvalues are distinct, matrices are diagonalizable (over the complex)

86 and the spectrum is the only invariant. The situation is very different in the skew-symmetric case and even in the simplest situations, the spectrum does not contain enough information to characterize the conjugacy class. This is the role of the signature. To give a very simple example, consider rotations R1,R2 in the plane R2 of angles α and −α. They are both symplectic automorphisms of R2 equipped with the standard symplectic form. They have the same spectrum exp ±2πiα. Of course, they are conjugate in GL(2, R) but not in SL(2, R): they don’t rotate in the same direction. The description of conjugacy classes of symplectic automorphisms is fundamental in hamiltonian dynamics since the flows under consideration are symplectic. For some reasons, dynamicists and topologists did not collabo- rate too much on these questions. The first (almost complete) solution to the problem has been given by Williamson [147] in 1936, with dynamical motivations. Then, a better understanding, still with dynamical motivations, was given in the wonderful book by Yakubovich and Starzhinskii [149]. See Eke- land [44], Long and Dong [89], Gutt [58] for recent and detailed presentations, also with dynamical motivations. From the point of view of algebraic topology, the best reference is another beautiful paper of Milnor [96] on isometries of inner product spaces, which deals with the general case over any field. We shall not give the full description and content ourselves with the part which is relevant for the definition of the signatures. Our presentation is only slightly different and uses basic ideas from Krein’s theory. From the symplectic form Ω on R2n, one produces canonically a hermitian form G on C2n. The hermitian product of u, v in C2n is given by: G(u, v) = iΩ(u, v). Note that all real vectors in R2n are isotropic for G. Also, note that any symplectic automorphism of (R2n, Ω) is also a unitary automorphism of (C2n,G). Given a complex subspace E of C2n one can consider the restriction of G to E. As any hermitian form has a signature, it follows that any subspace E has a signature τ(E). Of course complex conjugate subspaces have opposite signatures. Finally, let A be a symplectic automorphism (or more generally a uni- 2n tary automorphism of C ) and λ a complex number. Let Eλ(A) be the corresponding characteristic subspace: [ j Eλ(A) = ker(A − λId) j>1

87 (which is non trivial only if λ is an eigenvalue of A). The signature of the restriction of G to this subspace is called the λ-signature of A

λ τ (A) = τ(Eλ,G|) ∈ Z. In other words, we have a map

M λ M A ∈ Sp(2n, R) 7→ τ (A) ∈ Z λ λ∈C with each τ λ(A) ∈ Z a conjugacy invariant. The spectrum of symplectic A can be decomposed in disjoint groups (possibly empty) of four diﬀerent types.

−1 • 4-tuples {λ, λ−1, λ, λ },

• pairs of the form {λ, λ−1} for some real λ diﬀerent from ±1,

•± 1,

• pairs of the form {λ, λ−1} for some λ of modulus 1.

It is very easy to see that the corresponding τ λ(A) can be non trivial only in the case of a non real eigenvalue of modulus one. Summing up this discussion, we conclude that one can attach to each symplectic matrix A a function on the circle:

1 λ λ ∈ S 7→ τ (A) ∈ {−2n, . . . , 2n}.

λ Note that τ (A) = −τλ(A) so that one could assume that the imaginary part of λ is positive, without losing information. Note also that τ λ(A) = 0 if λ is not in the spectrum of A. When applied to the symplectic automorphism A attached to a knot k : S1 ,→ S3, one gets (a slight modiﬁcation of) Milnor’s signatures. For any 2πiθ1 2πiθ2 1 ω1 = e , ω2 ∈ e ∈ S with 0 6 θ1 < θ2 < 2π and a single eigenvalue 2πiµ λ = e with θ1 < µ < θ2 the jump in the ω-signatures is

σω2 (k) − σω1 (k) = σω2 (A) − σω1 (A) = 2τ λ(A) ∈ Z (Matumoto[91], Ranicki [118, Prop. 40.10]).

88 All the previously discussed signatures can be obtained from these τ λ. For instance, the Murasugi signature is the sum of all the τ λ for all λ with =(λ) > 0. Let us work out a simple example. Let cos α − sin α A = . sin α cos α The eigenvalues are exp(±iα) and the corresponding eigenvectors are (1, ∓i). Computing the G norm of these vectors, we get G((1, ∓i), (1, ∓i)) = iΩ((1, ∓i), (1, ±i)) = ±2 so that τ exp(±iα) = ±1.

3.4 Braids See Epple [45] for the history of braids in the 19th century, starting with Gauss: here is a braid drawn by him in 1833:

2 Fix n > 2 and n distinct points z1, z2, . . . , zn ∈ D . An n-strand braid b is an embedding a β : I = {1, 2, . . . , n} × I,→ D2 × I n

β(k,−) 2 projection I / D × I / I (1 6 k 6 n) is a homeomorphism, and

2 2 β(k, 0) = (zk, 0) ∈ D × {0}, β(k, 1) = (zσ(k), 1) ∈ D × {1} for some permutation σ of {1, 2, . . . , n}. Such a β deﬁnes n disjoint forward 2 3 paths t 7→ β(k, t) in D × I,→ R from (zk, 0) to (zσ(k), 1), such that each section β({1, 2, . . . , n} × I) ∩ (D2 × {t})(t ∈ I) consists of n points.

Braids in the Book of Durrow (Ireland)

Artin [10] founded the modern theory of braids. The trivial n-strand braid is

a 3 σ0 : ti ∈ I 7→ (i, ti, 0) ∈ R n

i i

i+1 i+1

90 For i = 1, 2, . . . , n − 1 the elementary n-strand braid σi is obtained from σ0 by introducing an overcrossing of the ith strand and the (i + 1)th strand, with the transposition (i, i + 1).

i i+1

i+1 i

−1 The inverse elementary n-strand braid σi is deﬁned in the same way but with an under crossing.

i i+1

i+1 i

The Artin group of isotopy classes of n-strand braids under concatenation is denoted by Bn - it has generators σ1, σ2, . . . , σn−1 and relations ( σiσj = σjσi if |i − j| > 2 σiσjσi = σjσiσj if |i − j| = 1.

Every n-strand braid β is represented by a word in Bn in ` generators, corresponding to a sequence of ` crossings in a plane projection. The closure of an n-strand braid β is the c-component link

a a 1 3 βb = β ∪ σ0 : I ∪σ I = Sc ,→ R n n with c = |{1, 2, . . . , n}/σ| the number of cycles in σ. Alexander [3] proved that every link is the closure βb of some braid β. For example, the Hopf link

91 i+1

i+1 i is the closure of the concatenation σ1σ1

1 1

2 2

The 2-strand braid β = σ 1 σ1

The closure β = Hopf link

An n-strand braid β with ` crossings is represented by a word in Bn of length ` in the generators σ1, σ2, . . . , σn−1, so that β = β1β2 . . . β` is the concatenation of ` elementary braids. Stallings [129] observed that the closure βb has a canonical projection with n Seifert circles and ` intersections, and hence a canonical Seifert surface with n 0-handles and ` 1-handles a 2 a 1 1 3 Fβ = D ∪ D × D ,→ R n ` and hence a canonical Seifert matrix Ψβ. This surface Fβ is homotopy equivalent to the CW complex n ` a 0 a 1 Xβ = ei ∪ ej i=1 j=1

1 0 0 with ∂ej = ei ∪ ei+1 if jth crossing is between strands i, i + 1

H1(Fβ) = H1(Xβ) = ker(d : C1(Xβ) → C0(Xβ)) = ker(d : Z` → Zn) = Zm.

92 We refer to Stallings [129], Gambaudo and Ghys [51], Cohen and van Wijk [36], Collins [37], Bourrigan [28] and Palmer [106] for more detailed accounts of the Seifert surfaces and Seifert matrices of braids.

Beard pullers and braids in the Book of Kells (Ireland)

3.5 Knots and signatures in higher dimensions There is also a high-dimensional knot theory, for knots (or links)

n n+2 k : S ,→ S in all dimensions n > 1. As noted by Adams [1] : Of course, from the point of view of the rest of mathematics, knots in higher-dimensional space deserve just as much attention as knots in 3-space.

93 The high-dimensional theory was initiated by Kervaire [78] using surgery methods, including the use of the plumbing construction to realize any bilinear pairing Σ : H × H → Z on a f.g. free Z-module H with (H, Σ + (−1)`Σ∗) a nonsingular (−1)`-symmetric form over Z as the Seifert form of a (2` − 1)- knot k : S2`−1 ,→ S2`+1 with an (` − 1)-connected Seifert surface F 2` ,→ S2`+1 such that H`(F ; Z) = H, for any ` > 1. The inﬁnite cyclic cover X is (` − 1)- connected; Levine [87] proved that for ` > 2 the isotopy classes of such simple (2` − 1)-knots k : S2`−1 ,→ S2`+1 are in one-one correspondence with the S-equivalence classes of Seifert matrices Σ with Σ + (−1)`Σ∗ invertible. n n+2 A cobordism of n-knots k0, k1 : S ,→ S is an embedding

n n+2 ` : S × I,→ S × I such that n `(u, i) = (ki(u), i)(u ∈ S , i ∈ {0, 1}). A knot k is slice if it extends to an embedding of the ball Dn+1 in the sphere Dn+3. The set of cobordism classes of n-knots is an abelian group Cn, with addition by connected sum.

Theorem 3.1 (Kervaire [78]). 1 1. For n > 2 every n-knot is cobordant to one with p : X → S a homotopy equivalence (resp. `-connected) if n = 2` (resp. n = 2` − 1). 2. C2` = 0 for ` > 1. 3. There are canonical maps C2`−1 → C2`+3 which are isomorphisms for ` > 2 and a surjection for ` = 1.

The knot cobordism groups C2`−1 for ` > 2 were computed modulo torsion by Levine [86]

M M M 0 C2`−1 = Z ⊕ Z4 ⊕ Z2 (countable ∞ s). ∞ ∞ ∞

2`−1 The high dimensional cobordism class k ∈ C2`−1 of a (2`−1)-knot k : S ,→ S2`+1 is determined modulo torsion by primary invariants of the signature type, with one Z-valued signature for each root eiθ ∈ S1 of the Alexander −1 polynomial ∆k(z) ∈ Z[z, z ] with 0 < θ < π. See Ranicki [118] for a recent account of high-dimensional knot theory, including the many applications of the signature in the computation of C2`−1

94 for ` > 2. For the relationship between knot invariants involving the infinite cyclic cover of the knot complement and Seifert surfaces see Blanchfield [22], Kearton [76] and Ranicki [119]. The computation of the knot cobordism groups is much more complicated for classical knots, i.e. for C1. Casson-Gordon [33] showed in 1975 that the surjection C1 → C4?+1 has non-trivial kernel, detected by the signatures of quadratic forms over cyclotomic fields. The kernel has been studied by Cochran, Orr and Teichner [35] using L2-signatures and noncommutative Blanchfield forms.

4 The symplectic group and the Maslov class

4.1 A very brief history of the symplectic group It is difficult, if not impossible, to describe the history of the symplectic group. Brouzet [30] mentions a double origin. The first is projective geometry during the nineteenth century, in particular connected with the concept of duality. Duality with respect to a quadric, i.e. a quadratic form, was generalized to duality with respect to a symplectic form, and this turned out to be relevant in the study of the “space of lines” and in particular in the study of “complexes” (families of lines in 3-space depending on two parameters). The second is related to algebraic geometry, in particular with the foundational work of Riemann on abelian integrals. Indeed, in modern terminology, the first cohomology of a Riemann surface is canonically equipped with a symplectic structure, defined by the cup-product, in duality with the intersection of cycles. Surprisingly, this paper does not mention a third origin, which may even be more important, at least from our point of view: the geometrization of mechanics all along the nineteenth century. One should mention many names. Among them Lagrange, Laplace, Legendre, Poisson, Hamilton, Li- ouville should be emphasized. Notice that they all have their lunar craters.

Towards the end of the nineteenth century, it was understood that the equations of motion of a mechanical system can be written in canonical Hamilton form: dp ∂H(p, q) dq ∂H(p, q) = ; = − dt ∂q dt ∂p

95 Lagrange Laplace Legendre

Hamilton Liouville Poincar´e

where q = (q1, . . . , qn) and p = (p1, . . . , pn) denote the positions and momenta and H is the hamiltonian function defined on R2n. This ordinary differential equation generates of flow φt of diffeomorphisms of R2n describing the motion. Even though one should probably not attribute a full credit to Poincaré,it seems to us that his famous memoir on the three body problem contains the first explicit formulation of the basic properties of symplectic geometry :

n t P • The ﬂow φ preserves the closed form Ω = dpi ∧ dqi. i=1

• It f is any diffeomorphism of R2n preserving Ω (a canonical transformation) then the flow associated to H ◦ f is f ◦ φt ◦ f −1. Using this, Poincaréfounded the qualitative theory of symplectic (or hamiltonian) dynamics. For instance, he used the preservation of the volume form Ωn and his recurrence theorem to prove some weak stability (“àla Poisson”) of the solar system. He also began a thorough analysis of periodic orbits. Taking the differential of φt at a fixed point, one gets what is called today a symplectic matrix and any conjugacy invariant of this matrix (for instance its

96 eigenvalues), gives some information on the dynamics. In particular, Poincar´e −1 knew that if λ is an eigenvalue of such a matrix, so are λ−1, λ, λ . He also used in many circumstances what we call today “symplectic manifolds”. The systematic study of the symplectic group, as a real algebraic group, began later. The terminology “symplectic” is due to Hermann Weyl in his 1939 book [146] on the classical groups.

The name “complex group” formerly advocated by me in allusion to line complexes, as they are deﬁned by the vanishing of antisym- metric forms, has become more and more embarrassing through collision with the word “complex” in the connotation of complex number. I therefore propose to replace it by the corresponding Greek adjective “symplectic”. Dickson calls the group “Abelian linear group” in homage to Abel who ﬁrst studied it.

The terminology “symplectic” has survived and fortunately the word “Abelian” is not used any more in this context since this would have created a big con- fusion! The entréeen scène of the symplectic group in topology came even later. Of course, the same Poincaréhad introduced homology and intersections of cycles in manifolds and “proved” his duality theorem but he does not seem to have used the symplectic property in any way. Recall that the homology for Poincaréwas not thought as a group. During the twentieth century, there was an incredible blooming of symplectic dynamics and topology, in particular thanks to the fundamental works of Arnold and the introduction by Gromov of the techniques of pseudo- holomorphic curves. The name of Arnold was given to the asteroid 10031 Vladarnolda, 2.5891829 astronomical units from the Sun, and whose orbit has a pretty large eccen- tricity 0.1991397.

97 Vladarnolda

4.2 The Maslov class The Maslov class appears in diﬀerent guises in many diﬀerent parts of mathematics.

• In mathematical physics, as originally introduced by Maslov in [90].

• In hamiltonian dynamics, for example though the introduction of the Conley-Zehnder index for periodic orbits.

• In topology where it is closely related to the signature of 4k-dimensional manifolds, the basic invariant of surgery theory.

• In algebra, speciﬁcally in the study of Witt groups, and their nonsimply-connected generalizations the Wall surgery obstruction groups.

• In number theory, since it is related to some automorphic forms.

There are excellent survey papers on the Maslov class, for example Cap- pell, Lee and Miller [32], de Gosson [39] and Py [110], providing a uniﬁed

98 presentation of the many facets of this object. The reader will ﬁnd a rather complete bibliography at the website [114]. Here, we shall limit ourselves to a small selection, relevant to the other papers in this volume. We can claim neither for originality nor for novelty and the best we can do is to quote the paper by Arnold on this topic [7]:

In such a classical area as Sturm’s theory it is hard to follow all the predecessors, and I can only say, like Bott and Edwards, that I do not make any claim as to the novelty of the results. In connection with this I remark that numerous authors writing on the Maslov index, symplectic geometry, geometric quantization, Lagrangian analysis, etc., starting with [2], have not noticed the earlier works of Lidskii, as well as the earlier works of Bott [3] and Edwards [4], in which was constructed a Hermitian version of the theory of the Maslov index and Sturm intersections.

Equip R2n with the standard symplectic form

n X Ω((xi), (yi)) = (xiyn+i − xi+nyi) ∈ R. i=1

A lagrangian is an n-dimensional subspace of Rn on which Ω vanishes identically. This terminology was introduced by Arnold [7] after Maslov had deﬁned in 1965 a “Lagrange manifold” as an n-dimensional submanifold of R2n in which all tangent spaces are lagrangian subspaces. Weinstein develops in [144] the credo that

Everything is a Lagrange manifold.

The space of lagrangian subspaces of R2n is a compact manifold tradi- tionally denoted by Λn. Of course, the symplectic group Sp(2n, R) acts on Λn and it is not hard to check that this action is transitive, so that Λn is a homogeneous space. The unitary group U(n) consists of complex unitary automorphisms of n n P 0 C equipped with the standard hermitian form zjz j. In the identiﬁca- j=1 tion between Cn and R2n, the imaginary part of the hermitian form is the symplectic form so that the unitary group U(n) is a (compact) subgroup of Sp(2n, R). It turns out that U(n) is indeed a maximal compact subgroup,

99 and that Λn is also homogenous under U(n) and is isomorphic to the quotient U(n)/O(n). The homogenous space Hn = Sp(2n, R)/U(n) is called the Siegel domain and carries very interesting geometry. This is a symmetric space with non positive curvature. This is also a complex holomorphic domain, that one can also see as the set of complex symmetric n × n matrices chose imaginary part is definite positive. Siegel devoted a long paper to this geometry in 1943 [127]. His motivation came from number theory and the generalization of modular forms. It is always a good idea to look at the case n = 1 where Sp(2, R) be- comes SL(2, R), the lagrangian Grassmannian Λ1 is the real projective line (diffeomorphic to a circle), and the Siegel space is the Poincaréupper half space. The Maslov class may be seen in many related ways.

1 • An element of H (ΛN , Z),

• A cohomology class of degree 2 of Sp(2n, R) deﬁning its universal cover.

• A two cocycle on Λn invariant under the action of Sp(2n, R).

• A two cocycle on Hn invariant under the action of Sp(2n, R).

We shall now describe these objects.

4.3 A short recollection of group cohomology Let Γ be any group, equipped with the discrete topology. On can consider the simplicial complex whose vertices are the elements of Γ and having a k-simplex for every (k + 1)-tuple of elements of Γ. This space is obviously contractible and carries an obvious free action by left translation of Γ on itself. Clearly, the quotient space is a model for the Eilenberg MacLane space K(Γ, 1). The cohomology of this quotient space with coeﬃcients in some abelian group A is called the group cohomology of Γ and denoted by H?(Γ,A). It is computed from the diﬀerential graded algebra consisting of cochains c :Γk+1 → A which are invariant under translations :

c(γγ0, . . . , γγk) = c(γ0, . . . , γk).

100 These cochains are called homogeneous for obvious reasons. Every homogeneous cochain c deﬁnes a non-homogeneous cochain c :Γk → A by

c(g1, g2, . . . , gk) = c(1, g1, g1g2, . . . , g1g2 . . . , gk) and conversely any non-homogeneous cochain deﬁnes a unique homogeneous cochain. Therefore, one can compute the cohomology of a group using non- homogeneous cochains. For instance, in the non-homogeneous presentation, a 1-cochain is just a map f :Γ → A and its coboundary is the map

2 (g1, g2) ∈ Γ 7→ df(g1, g2) = f(g1g2) − f(g1) − f(g2).

Cohomology is typically used to describe central extensions

i π {0} / A / Γe / Γ / {1} .

One chooses any set theoretic section s from Γ to Γe (i.e. such that π ◦ s = −1 −1 Id). Then, given two elements g1, g2 in Γ, the element s(g1g2)s(g2) s(g1) projects by π to the identity so that it is an element of the kernel of i and can be identified with some element of A. In this way, we get a function c depending on g1, g2 with values in the abelian group A. This turns out to be a cocycle and changing the section s changes c by a coboundary so that the cohomology class of c is well defined. This is a class in H2(Γ,A) associated to the central extension. Conversely, a cohomology class of degree 2 defines a central extension. For an excellent introduction to group cohomology, we recommend [31].

4.4 The topology of Λn Since the determinant of a matrix in O(n) is ±1 and the determinant of a unitary matrix has modulus 1, the homogeneous space Λn = U(n)/O(n) maps to the circle:

1 2 p : U(n)/O(n) → S ; A 7→ det(A ). This is a locally trivial ﬁbration with simply connected ﬁbres SU(n)/SO(n). As a consequence, the fundamental group of Λn is isomorphic to Z.

101 As a first definition, the Maslov class is the corresponding element of 1 H (Λn, Z). Any loop in Λn has a Maslov index, the index of its projection by p. Note that the symplectic group Sp(2n, R) acts canonically on Λn but this action does not preserve the fibres of p. However, we shall see later that the action “almost preserves” the fibres. Since U(n) is a maximal compact subgroup of Sp(2n, R), the quotient Hn = Sp(2n, R)/U(n) is contractible. It follows that the fundamental group of Sp(2n, R) is also isomorphic to Z so that its universal cover Sp(2f n, R) is a central extension

i π {0} / Z / Sp(2f n, R) / Sp(2n, R) / {1} .

See Rawnsley [121] for an explicit description of Sp(2f n, R). The Maslov class is the cohomology class in H2(Sp(2n, R), Z) corresponding to this central extension. Another (equivalent) way to understand this class is to use the fact (proved by Hopf) that any homology class of degree 2 is represented by a surface. Let φ : π1(Σg) → Sp(2n, R) be some homomorphism from the fundamental group of a compact oriented surface of genus g to the symplectic group. The group π1(Σg) has a well known presentation

ha1, . . . , ag, . . . , b1, . . . bg | [a1, b1] ... [ag, bg] = 1i.

Choose elements φ](ai), φ](bi) in Sp(2f n, R) lifting φ(ai), φ(bi). The product of commutators [φ](a1), φ](b1)] ... [φg(ag), φ](bg)] is an integer independent of all the choices. This is the Maslov class evaluated on (the homology class represented by) φ.

4.5 Maslov indices and the universal cover of Λn Let us begin with the simple but crucial observation of Leray [85]. Suppose 2n L1,L2 are two transverse lagrangian subspaces of R . The pairing Ω : L1 × ? L2 → R is nonsingular so that L2 is identified with the dual L1 of L1. In this 2n ? way, the space R is identified with L1 ⊕ L1 and under this identification ? the symplectic structure on L1 ⊕ L1 is simply given by:

? ? ? ? Ω((x, x ), (y, y )) = y (x) − x (y) ∈ R.

102 Suppose now that we are given a third lagrangian L3 which is transverse to ? L2 so that it is the graph of some linear map l from L1 to L1. The fact that L3 is lagrangian means that l is a symmetric form, i.e. l(x)(y) − l(y)(x) ≡ 0. In other words, if L1 and L3 are transverse to L2, the lagrangian space L1 is canonically equipped with a quadratic form. One defines the Wall-Maslov ternary index W (L1,L2,L3) as the signature of this quadratic form. If turns out that this definition can be generalized without any transver- sality conditions. One of the possibilities is the following. As in Defini- n 2n tion 2.23 2., given three lagrangians L1,L2,L3 in H−(R ) = (R , Ω) one considers the vector space

2n V = {(v1, v2, v3) ∈ L1 ⊕ L2 ⊕ L3 | v1 + v2 + v3 = 0 ∈ R }, and the canonical quadratic form on V deﬁned by

QL1,L2,L3 (v1, v2, v3) = Ω(v1, v2) = Ω(v2, v3) = Ω(v3, v1).

One deﬁnes the Wall-Maslov triple index in general as the signature of this quadratic form. So far, we worked with the symplectic group over the real numbers. This construction can however be generalized over any ﬁeld K. The symplectic group Sp(2n, K) now acts on the Grassmannian Λn(K) of lagrangians in n H−(K ) and we can associate a quadratic form to three lagrangians, exactly as above. The Wall-Maslov index of the triformation (L1,L2,L3) is now a class of in the symmetric Witt group W (K). Let us give the main properties of this index:

3 • The Wall-Maslov ternary index W :Λn(K) → W (K) is invariant under the action of Sp(2n, K).

• W is an alternating function of its three arguments (L1,L2,L3).

• W is a cocycle : for any four lagrangians L1,L2,L3,L4, one has

W (L2,L3,L4) − W (L1,L3,L4) + W (L1,L2,L4) − W (L1,L2,L3) = 0

in W (K).

Only the third property is a tricky exercise in linear algebra. See Py [110] for a short proof.

103 Let us come back to the case of real numbers, which is our main example. Since the fundamental group of Λn is isomorphic to Z, its universal cover π : Λen → Λn is an infinite cyclic covering space of Λn. Let us denote by T : Λen → Λen the generator of the deck transformations. The fibration 1 p :Λn → S lifts to a fibration pe : Λen → R such that pe(T (Le)) = pe(Le) + 1. Of course, the group Sp(2f n, R) acts on Λen. Several important properties should be mentioned. The first concerns the topology of Λn. If one fixes a lagrangian L, one can consider the space Λn,tL consisting of lagrangians transverse to L. As noticed above, this space is identified with the space of quadratic forms on any lagrangian supplementary space of L. More precisely, Λn,tL is an affine space modeled on the space of symmetric matrices : the “difference” between two elements of Λn,tL being a quadratic form. This provides an atlas for Λn. The complement of Λn,tL in Λn is called by Arnold the train of L. The 3-cocycle W is a coboundary when lifted to Λen. This means that 2 there is a function m : Λen → Z such that for any three elements of Λen, we have:

m(Le1, Le2) + m(Le2, Le3) + m(Le3, Le1) = W (π(Le1), π(Le2), π(Le3)).

Moreover, this binary index is unique if one imposes the conditions

m(L,e T (Le)) = n and the invariance under the action of Sp(2f n, R). The existence and unique- ness of this function m is explained in a crystal clear way in Arnold [7]. Suppose Lt is a continuous path of lagrangians which are all transversal to the same L. Then the “difference” of L1 and L2 define a quadratic form having some signature. One then lifts the path to a path Let in Λen and one defines m(Le1, Le0) as being this signature. One can checks that this is independent of the path connecting these two lagrangians. This only defines m in the very special case of two elements of Λen which are connected by a path everywhere transverse to a given lagrangian. One can then extend the definition of m to all pairs of elements of Λen by using the required property that M is the coboundary of m. The second property is that Λen is canonically equipped with a partial ordering which is important in hamiltonian dynamics. Any Λn,tL, being contractible, can be lifted to countably many contractible sets in Λen. In

104 each if these contractible sets, on defines Lf1 6 Le2 if the difference is a non negative quadratic form. One should check that these definitions are indeed compatible in the intersections of the Λn,tL. Then one defines in general Lf1 6 Le2 if there is a chain of elements of elements of Λen connecting them, so that two consecutive elements of the chain belong to one of these contractible parts, and forming an ascending chain for the partially defined ordering. These verifications are not difficult.

4.6 The Maslov class as a bounded cohomology class One of the key properties of the Maslov class is that it is bounded. From the na¨ıve point of view, this simply means that the Maslov triple index takes values in a finite set {−n, . . . , n}. Given a space X, its singular homology is defined using singular simplices, i.e. continuous maps τ from the standard simplex ∆k in X. The space Ck(X) of singular chains is by definition the vector space generated by k simplices. One can equip this space with the `1-norm, i.e. the sum of the absolute values of the coefficients. Singular cochains are defined as the dual of singular chains. A cochain is called bounded if it is a continuous linear form, with respect of the `1-norm. More concretely, a cochain c is bounded if there is a constant C such that the evaluation c(τ) on any k-simplex has absolute value less that C. The vector space of bounded cochains is obviously a graded differential complex. Its cohomology is called the bounded cohomology of X and denoted ? by Hb (X). This cohomology was introduced by Gromov [56]. As a typical example, consider a compact oriented surface Σ of genus g > 2 so that it can be equipped with a riemannian metric of curvature −1 and that one can identify its universal cover with Poincarédisc. Consider a 2-simplex τ : ∆2 → Σ. Lift it to the Poincarédisc as a simplex τe and define c(τ) as the (oriented) area of the geodesic triangle having the same vertices as τe. One checks easily that this is defines a cocycle c which is bounded since the area of geodesic triangles in the Poincarédisc is less than π by Gauss-Bonnet theorem. The cohomology class of this 2-cocycle in the usual cohomology of Σ is easy to compute : this is the area of Σ. This example is somehow typical and bounded cohomology is vaguely related to negative curvature. Given a space X, there is a classifying map to the Eilenberg MacLane space K(π1(X), 1). Gromov showed that this map induces an isomorphism

105 in bounded cohomology. In particular, simply connected spaces have a trivial bounded cohomology. Therefore, all information concerning bounded cohomology comes from the fundamental group and should be computed from the group theoretical bounded cohomology. Let Γ be any group, equipped with the discrete topology. One considers the diﬀerential graded algebra consisting of bounded homogeneous cochains c :Γk+1 → R. Its homology is the bounded cohomology ? of the group Γ and denoted by Hb (Γ, R). Let us state some of the main properties of this cohomology, especially in degree 2. See Grigorchuk [55] for a survey of the second bounded cohomology of a group. If Γ is amenable (for example if it is solvable) all bounded cohomology groups vanish. If a group Γ is uniformly perfect, i.e. if there is an integer k such that any element is a product of at most k commutators, then the canonical map 2 2 from Hb (Γ, R) to H (Γ, R) is injective. Indeed, suppose a bounded 2-cocycle is the coboundary of some a priori unbounded 1-cochain f :Γ → R. This means that there is a constant C > 0 such that for every g1, g2 one has

|f(g1g2) − f(g1) − f(g2)| 6 C. Such maps f are called quasi homomorphisms. It is easy to see that a quasi- homomorphisms is uniformly bounded on the set of products of a given number of commutators. Hence, the assumption that Γ is uniquely perfect im- 2 plies that f is bounded, which means precisely that the mapping Hb (Γ, R) to H(Γ, R) is injective. 2 2 In general, the kernel of Hb (Γ, R) to H (Γ, R) is described by quasi- homomorphisms up to bounded functions. There is an elementary way to get rid of these bounded functions using “homogenization”. Starting from a quasi homomorphism f, one deﬁnes its homogenization by

f(gk) f(g) = lim k→∞ k which is another quasi-homomorphism such that |f − f| is bounded. In this way, one sees that the kernel under consideration is identiﬁed with homogeneous quasi-homomorphisms, i.e. satisfying the additional condition that f(gk) = kf(g). It turns out that the symplectic group Sp(2n, R) is uniformly perfect.

106 The Maslov class is a typical example of a bounded class. Chose a lagrangian L in Λn. If g0, g1, g2 are three elements of Sp(2n, R), the ternary index W (g0(L), g1(L), g2(L)) deﬁnes a bounded 2-cocycle since its values are bounded by 2n. This is the bounded Maslov class. Clearly, the pull back this cocycle to Sp(2f n, Z) is exact and is the coboundary of the 1-cochain associating to (ge0, ge1) the binary index m(ge0(Le), ge1(Le)). The Maslov class in H2(Sp(2n, R), Z) is associated to the central extension

i π {0} / Z / Sp(2f n, R) / Sp(2n, R) / {1} . Of course, the above definition of the Maslov cocycle depends on the choice of a lagrangian L, as a base point in Λn, but the class does not. In order to define a canonical cocycle, one can proceed in two ways, leading to two interesting concepts. In a first approach, one can pull back some bounded Maslov cocycle, associated to some choice of L, to Sp(2f n, R) where, as we noticed already, it is exact. Therefore, there is a map r : Sp(2f n, R) → R such that

−1 • t(ge1ge2) )t(ge1)t(ge2) = W (L, π(ge1(L)), π(ge2(L))

• t(T ge) = t(ge) + 1. This function t is of course a quasi-homomorphism which depends on L but its homogenization ρ does not. The function trans : Sp(2f n, R) → R is called the symplectic translation number. This is the unique homogeneous quasi-homomorphism trans : Sp(2f n, R) → R taking the value 1 on the generator T of the center. Modulo Z, one gets the rotation number ρ on the symplectic group, with values in R/Z, useful in dynamics. In a second approach, one can double the dimension of R2n and consider R2n ⊕ R2n with the symplectic form Ω ⊕ −Ω. The graph of a symplectic 2n 2n automorphism g : H−(R ) → H−(R ) is a lagrangian

2n Γg = {(x, g(x)) | x ∈ R }.

2n 2n Three elements g0, g1, g2 of Sp(2n, R) deﬁne three lagrangians in R ⊕ R , for which one can compute the Wall-Maslov ternary index. This deﬁnes a Maslov 2-cocycle on the symplectic group which is canonical in the sense that it is invariant under conjugacy and does not depend on the choice of a lagrangian.

107 Finally, we mention another geometrical interpretation of the Maslov class. The Siegel domain Hn = Sp(2n, R)/U(n) is a Kähler manifold whose curvature is non positive. It can be compactified in an equivariant way by adding a sphere at infinity. Contrary to the case of the Poincarédisc (when n = 1), the action of Sp(2n, R) on this sphere is not transitive. The structure of this heterogeneous sphere is well understood and is described by a Tits building, that we cannot describe here. It suffices to say that one piece of this building is identified with Λn which is therefore equivariantly embedded in the sphere at infinity. Now consider three points x, y, z in Hn. One can connect them by three geodesic arcs [x, y], [y, z], [z, x] to build a triangle. However, there is no to- tally geodesic plane containing the three points so that one cannot construct a usual triangle having x, y, z as vertices. Nevertheless, one can choose any smooth triangle having the union of [x, y], [y, z], [z, x] as its boundary and in- tegrate the Kählerform on this triangle. Since the form is closed, this “area” does not depend on the choice of the triangle so that one can indeed define some number area(x, y, z) ∈ R. This is a bounded cocycle. Choosing some base point ? in Hn one gets 2-cocycles in Sp(2n, R) as area(?, g1(?), g2(?)). Changing the base point does not change the cohomology class, as one checks easily. When ? goes to the sphere at infinity and converges to some lagrangian, the corresponding cocycle converges to the previously defined Maslov cocycle. In other words, the Maslov class can also be seen as some incarnation of the symplectic form of Siegel space. Suppose now that we restrict our study to integral symplectic matri- −1 −1 ces in Sp(2n, Z). Realizing g0 g1, g0 g1 ∈ Sp(2n, Z) by automorphisms α, β :Σn → Σn there is defined a 4-dimensional manifold with boundary (T (α, β), ∂T (α, β)) such that

1. the double mapping torus T (α, β) is a ﬁbre bundle Σn → T (α, β) → P over the 2-dimensional pair of pants P with ∂P = S1 t S1 t S1, 2. the mapping torus

T (α) = Σn × [0, 1]/{(x, 0) ∼ (α(x), 1) | x ∈ Σn}

1 is a ﬁbre bundle Σn → T (α) → S with monodromy α, such that ∂T (α, β) = T (α) t T (β) t T (α ◦ β).

108 The signature of T (α, β) is the Wall non-additivity invariant (2.23)

Meyer(g0, g1, g2) = τ(T (α, β)) n n n = W (H1(Σn; R); R ⊕ 0, α(R ⊕ 0), β(R ⊕ 0)) 2n = W (H−(R ); Γg0 , Γg1 , Γg2 ) ∈ Z which is called the Meyer cocycle on Sp(2n, R) and which is clearly independent of all choices - this is a very useful cocycle in topology.

4.7 Picture in the case n = 2

The case n = 1 is well known. Λ1 is the projective line over the reals, so that it is a circle. The symplectic group in this case is SL(2, R) and acts on Λ1 through its quotient PSL(2, R) in the well known way. The Siegel domain H1 is the Poincarédisc. Any element of PSL(2, R) can be seen as an orientation preserving homeomorphism of the circle, having a therefore a well defined rotation number in R/Z. It might be useful to draw a picture in the case n = 2 which clarify the situation. We follow the remarkable paper by Arnold [8]. Let us consider Λ2 = U(2)/O(2). It fibres over the circle with fibres diffeomorphic to SU(2)/SO(2), which is diffeomorphic to the 2-sphere. The 2 universal cover Λe2 is therefore diffeomorphic to S ×R. The second coordinate is the lift of the map det2. It is not difficult to check that the generator T of the deck transformations acts as T (x, t) = (−x, t + 1) so that one should be careful that Λ2 is not orientable. If one chooses some lagrangian L, its train, that is the set of lagrangians which are not transverse to L is homeomorphic to a 2-sphere in which two antipodal points have been identified. Lifting this 2 train to Λe2 = S × R, one gets a string of two spheres, glued along points. The complement of this “rosary” consists of disjoint copies of Λ2,tL, each being identified to the space of quadratic forms in R2, i.e. to the 3-space R3. 2 The following picture shows Λe2 = S ×R. The grey part represents Λ2,tL. The three local pictures represent the situation in the neighborhood of the three singular points (x, t), (−x, t + 1), (x, t + 2).

109 The spheres S2 × {t} are deﬁnitely not invariant by the symplectic group Sp(4f , R). However, it follows from the boundedness of the Maslov class that 2 if one looks at the image ge(S × {t}) of such a sphere by an element of Sp(4f , R), its projection on the R factor is an interval of bounded length. In other words, even if the action of Sp(4f , R) does not descend to an action on R, it “quasi-descends” if one accepts to deﬁne a “quasi-action” in such a way that the image of a point might be an interval of bounded length. Concerning the Siegel domain H2, its dimension is 6, so that its complex dimension is 3. Its geometrical boundary is a 5-sphere S5 that we now describe in terms of the Tits building associated to this situation. 5 As mentioned earlier, S contains a copy of Λ2. It also contains a copy of 3 RP , the projective space of lines in R4. The sphere S5 is a “restricted link” 3 of RP and Λ2. This means that every point on the sphere can be considered as a “barycenter” λD + (1 − λ)L of some line D and some lagrangian plane L containing D for some λ ∈ [0, 1].

110 5 Dynamics

In this section, we brieﬂy describe three topics from dynamical systems where some “Maslov type” ideas are relevant.

5.1 The Calabi and Ruelle invariant Let us explain the construction of some invariants of symplectic diﬀeomor- phisms in the simplest case. Let B be the unit open ball in R2n and denote by Sympc(B) the group of symplectic diﬀeomorphisms of B with compact support.

There are two basic dynamical invariants for elements of Sympc(B). Choose some primitive for the symplectic form Ω, for instance X λ = pidqi.

? Let φ be an element of Sympc(B) so that d(φ (λ) − λ) = 0. It follows that there is a function H : B → R such that dH = φ?(λ) − λ and which is uniquely deﬁned if one imposes the condition that H is zero near the boundary of B. One deﬁnes the Calabi invariant of φ as the integral of H: Z C(φ) = H Ωn. B

It is not hard to see that C is a homomorphism from Sympc(B) to R. It is much harder to prove that the kernel of C is a simple group, so that every non injective homomorphisms from Sympc(B) to a group factors through C. See Banyaga [15] for details. Except in dimension 2, the dynamical meaning of this homomorphism is unclear. In dimension 2, it is related to some kind of linking between orbits (see for instance Gambaudo and Ghys [49]). The Ruelle invariant is related to the Maslov index (see Ruelle [122]), and has been discussed in Barge and Ghys [16]. Let φ be an element of Sympc(B) and consider its diﬀerential:

dφ : x ∈ B 7→ dφ(x) ∈ Sp(2n, R).

111 One can lift this to Sp(2f n, R) as a map

dφf : x ∈ B 7→ dφf(x) ∈ Sp(2f n, R) which is the identity near the boundary of B. Therefore one can compute the symplectic translation number t(dφf(x)). Its integral over B defines a quasi-homomorphism on Sympc(B) whose homogenization is called the Ru- elle invariant R: Z 1 k n R : φ ∈ Sympc(B) 7→ lim t(dφg(x)) Ω ∈ R. k→∞ k B Somehow, one can describe this invariant as an “elliptic Lyapunov expo- nent”. Recall that Lyapunov exponents describe the exponential growth of a vector under iteration of a diffeomorphism. In our case, the symplectic translation number only takes into account the eigenvalues of modulus one of the differential. As a very simple example, one can consider the case n = 1 and the p 2 2 hamiltonian function H : B → R which is of the form h( p1 + q1) for some real function h : [0, 1] → R which is constant in a neighborhood of 0 and which is 0 in a neighborhood of 1. Let φ be the time one of the hamiltonian flow associated to H. One can easily compute the invariants of Calabi and Ruelle and see that they are different: Z 1 C(φ) = 2πrh(r) dr and R(φ) = −h(0). 0 See [16, 49] for more information about these invariants. Let us simply mention that similar invariants can also be defined on the groups of symplec- tomorphisms of more general symplectic manifolds. For the state of the art on this topic, we recommend Borman and Zapolsky [25] or Shelukhin [126]. Some of these invariants are related to Floer homology.

5.2 The Conley-Zehnder index Recall that a symplectic manifold is an even dimensional manifold M equipped with a closed non degenerate 2 form Ω. A function H : M → R on such a manifold defines a vector field XH called its symplectic gradient such that t Ω(XH , −) = dH(−). The flow generated by XH is the hamiltonian flow φH associated to H. As a matter of fact, one usually considers hamiltonians Ht

112 depending periodically on a time t, i.e. from M × R/Z to R. One therefore has a vector field depending periodically on time generating some non au- t 1 tonomous flow φH . The symplectic diffeomorphism Φ = φH is by definition a hamiltonian diffeomorphism. As it is well known, at least since Poincaré,a great deal of information about the dynamics is contained in periodic orbits of Φ and it has been a constant problem to find tools proving the existence of these orbits. As Poincaréwrote in the “Méthodes Nouvelles de la MécaniqueCéleste”[108, page 82]:

Il semble d’abord que [l’existence de solutions périodiques] ne puisse être d’aucun intérêtpour la pratique. En effet, il y a une proba- biliténulle pour que les conditions initiales du mouvement soient précisémentcelles qui correspondent àune solution périodique. Mais il peut arriver qu’elles en diffèrent trèspeu, et cela a lieu justement dans les cas oùles méthodes anciennes ne sont plus ap- plicables. On peut alors avec avantage prendre la solution périodique comme première approximation [... ] D’ailleurs, ce qui rend ces solutions périodiques aussi précieuses,c’est qu’elles sont, pour ainsi dire, la seule brèchepar oùnous puissions essayer de pénétrer dans une place jusqu’ici réputéeinabordable.

Assume for simplicity that the manifold M is R2n and consider a periodic point x of period 1, so that Φ(x) = x. Taking the diﬀerential along the orbit, one gets a path

t t ∈ [0, 1] 7→ γ(t) = dφH (x) ∈ Sp(2n, R). Even though the periodic orbit is a closed loop, the above path is of course not necessarily a loop. We know how to associate a Maslov index to a loop in the symplectic group Sp(2n, R). Conley and Zehnder defined an index, in the same spirit as Maslov for a path γ(t) in the symplectic group such that γ(0) = I2n and γ(1) does not have 1 as an eigenvalue (which is a generic condition). See for instance [39] for a definition. In this way, every fixed point of a hamiltonian diffeomorphism has a Conley Zehnder index. This can be generalized in many symplectic manifolds. This index has been remarkably useful in the context of Floer homology, which is the most powerful tool to construct periodic orbits. It is impossible

113 to give any detail of this theory here and we refer to Audin and Damian [13] for a good exposition. Let M be a closed symplectic manifold. Assume for simplicity that M is 2-connected. Denote by LM the space of loops, i.e. the space of smooth maps from R/Z to M. Let Ht be a time periodic hamiltonian. We define the action functional A : LM → R in the following way. If γ : R/Z → M is a loop, choose a disc D with boundary γ and let Z Z A(γ) = − Ω − Ht(γ(t))dt. D R/Z It is easy to see that the critical points of this functional are precisely the 1-periodic points we are trying to detect. It is therefore tempting to define some Morse type homology on this loop space, which is called the Floer homology. One introduces a generic almost complex structure compatible with the symplectic form. This enables us to define the gradient lines in the loop space. Given two critical points, i.e. two periodic orbits γ+, γ−, one can look at the space of gradient lines γs∈R with γ+ such that γs converges to γ+ (resp. γ−) when s goes to +∞ (resp. −∞). In classical Morse theory, the dimension of the intersections of stable and unstable manifolds of two critical points of generic Morse functions is completely determined by the difference of the indices of the critical points. In this new context of Floer homology, indices are replaced by the Conley-Zehnder indices. Again, we refer to [13] for details.

5.3 Knots and dynamics In this subsection, we describe some analogies between the topology of knots and links and the dynamics of volume preserving flows on 3-dimensional manifolds, especially on the 3-sphere. More details can be found in Ghys [53]. Recall that if G is a Lie algebra, one can define a canonical Killing symmetric form by hx, yi = trace(ad(x)ad(y)). If G is a simple Lie algebra, this is nonsingular and this is the unique bilinear form which is invariant under the adjoint action (up to a constant factor). In some cases, infinite dimensional Lie algebras possess such a Killing form. The main example is the Lie algebra of divergence free vector fields on the 3-sphere, or more generally on a rational homology sphere Σ of dimension

114 3. Let vol be a volume form on Σ (of total mass 1) and consider the Lie algebra sdiff(M) of divergence free vector ﬁelds X on M, i.e. such that the Lie derivative LX vol = 0. This condition is equivalent to the fact that the 2-form iX vol is closed and hence exact, because of our assumptions. Let αX be a choice of a primitive. Then one can deﬁne Z

hX1,X2i = αX1 ∧ d αX2 . Σ This is a symmetric nonsingular bilinear form on sdiff(Σ) which is of course invariant under conjugacy. In particular, hX,Xi is an invariant of a divergence free vector ﬁeld, that we shall call the Arnold invariant, following Arnold’s Principle:

If a notion bears a personal name, then this is not the name of the discoverer. and its complement, the Berry Principle:

The Arnold Principle is applicable to itself ([6]).

Indeed, the invariant was introduced by many authors, including Arnold (and Moﬀatt, Morau), under the name of Hopf invariant. It is also called helicity. Arnold gave a nice dynamical or topological interpretation of hX1,X2i. t t Choose some auxiliary riemannian metric on Σ. Let φ1, φ2 be the ﬂows generated by X1 and X2 and pick two points x1, x2. One can follow the piece t1 of trajectory of X1 from x1 to φ1 (x1) and then follow some minimal geodesic connecting the two endpoints of this arc. In this way, we produce a closed loop kX1 (x1, t1). In the same way, we can consider the loop kX2 (x2, t2). It turns out that for a generic choice of x1, x2, t1, t2 these two loops are disjoint so that one can compute their linking number. Then, a form of the ergodic theorem shows that the limit 1 lk(x1, x2) = lim lk(kX1 (x1, t1), kX2 (x2, t2)) t1,t2→+∞ t1t2 exists for almost every pair of points (x1, x2). The dynamical interpretation is then that the Arnold invariant is an asymptotic linking number: Z Z hX1,X2i = lk(x1, x2) dvol(x1) dvol(x2). Σ Σ

115 The main open conjecture in this topic concerns the topological invari- t t ance. Suppose that the two flows φ1, φ2 are conjugate by some volume preserving homeomorphism. Does that imply that the Arnold invariants of X1 and X2 are equal? A partial result in this direction is obtained in Gambaudo and Ghys [49]. If the manifold Σ contains an invariant solid torus D × S1 in such a way that the fibres D × {?} are transversal to the vector field X, one can relate the contribution of helicity in the solid torus to the Calabi invariant of the first return map on a disc. Since it is known that the Calabi invariant in dimension 2 is invariant by area preserving homeomorphisms, this provides a wide class of vector fields for which the conjecture is true. The idea of considering a divergence free vector field as some kind of diffuse knot is certainly not new. The closed form iX vol can be considered as a closed 1-current. It is therefore tempting to try to generalize to volume preserving vector fields some other concepts coming from knot theory. In particular, in Gam- baudo and Ghys [50], we discussed the signatures of a divergence free vector field. As before, we consider the loop kX (x, t). For almost every choice of x, t, this is a knot and one shows that the limit of signatures 1 τX (x) = lim τ(kX (x, t)) t→+∞ t2 exists for almost every point x. Moreover, we show that if the vector field is ergodic, this limit τX (x) is almost everywhere constant and coincides with (one half of the) helicity of X. A wide generalization is obtained in Baader and Marché[14] to invariants of finite type in knot theory. They show, assuming ergodicity, that if v is a finite type invariant of order n, the limit 1 lim v(kX (x, t)) t→+∞ tn

n exists for almost every point x and is equal to CvhX,Xi for some constant Cv depending only on the invariant v.

116 6 Number theory, topology and signatures

6.1 The modular group SL(2, Z) The modular group SL(2, Z) is the group of 2 × 2 integer matrices a b A = c d such that det(A) = ad − bc = 1 ∈ Z. Implicitly, the modular group was introduced by Gauss in his famous Dis- quisitiones arithmeticae (1801), discussing in particular quadratic forms on two variables with integral coefficients. It has been thoroughly investigated by many mathematicians, starting with Dedekind [41] in his commentary on Riemann’s work on elliptic functions. This group is related to almost every area of mathematics and at least one of the authors of the present paper considers it as “his favorite group”. See Serre [124] or Mumford, Series and Wright [101] for a modern account. In this final section, we will restrict ourselves to a very specific aspect of the modular group: its relation with low dimensional topology and signatures. The center of SL(2, Z) consists of {±I2} and the quotient by the center is denoted by PSL(2, Z). The fundamental object is the action of PSL(2, Z) on Poincaréupper half plane H = {z ∈ C|=(z) > 0}. a b az + b ( , z) ∈ PSL(2, ) × H 7→ ∈ H c d Z cz + d (we use the bracket notation for the quotient by the center). This action preserves the hyperbolic metric and there is a well-known fundamental domain: 1 1 {z ∈ H| − <(z) and |z| 1}. 2 6 6 2 > The action is not free but the quotient X = H/PSL(2, Z) defines a Riemann surface which is isomorphic to C. This isomorphism is realized by the famous j-invariant: 1 j(z) = + 744 + 196884q + 21493760q2 + 864299970q3 + 20245856256q4 + ··· q

117 with q = exp(2πiz). However, it is frequently useful to think of the X = H/PSL(2, Z) as an orbifold, taking into account the two non-free orbits, with stabilizers Z/2Z and Z/3Z. The universal cover of X, as an orbifold, is H and its fundamental group, again as an orbifold, is PSL(2, Z). The following ﬁgure is extracted from a famous paper by Klein [81], dated 1878, in which he explains that this tessellation is due to Dedekind, who himself acknowledges the inﬂuence of Gauss...

The action on the boundary of H is the natural action of PSL(2, Z) on RP1 = R ∪ {∞}. The key point is that this action is transitive on rational numbers. For A ∈ SL(2, Z) with c 6= 0 the Euclidean algorithm gives a n regular χ ∈ Z with |χk| > 2, such that a b 0 −1 χ −1 χ −1 χ −1 A = = 1 2 ... n c d 1 0 1 0 1 0 1 0

118 1 a/c = [χ , χ , . . . , χ ] = χ − . 1 2 n 1 1 χ2 − .. χ3 − . 1 − χn This is essentially the same matrix product as in Section 1.6. One can express the same fact in several equivalent ways. It one sets

0 −1 1 1 1 1 S = ; T = ; and U = TS = , 1 0 0 1 1 0

2 3 one has S = U = I2 and PSL(2, Z) is the free product of the corresponding subgroups of order 2 and 3. One could also say that any element of PSL(2, Z) can be written in a unique way as a product of matrices which are alternatively of the form

1 n 1 0 and 0 1 n 1 with n 6= 0. Modular forms of weight 2k are expressions of the type f(z) dzk, with f holomorphic on H, which are invariant under PSL(2, Z). For an introduction, we recommend the wonderful [124]. We will only recall that there is essentially a unique modular form of weight 12,

∞ Y ∆(z) = q (1 − qn)24 n=1 again with q = exp(2πiz). To explain the existence of this ∆, one could use the kernel Γ2 of the abelianization of PSL(2, Z), of index 6, acting freely on H. The quotient of H by Γ2 is a bona ﬁde Riemann surface, covering 6 times X, which is actually a punctured elliptic curve, and which therefore carries a non vanishing holomorphic form. The sixth power of this holomorphic form goes down to the modular form ∆ on X.

6.2 Torus bundles and their signatures There are basically two ways of constructing a 3-manifold from a matrix in SL(2, Z). In this subsection, we describe torus bundles.

119 Every element A ∈ SL(2, Z) is induced by an automorphism of the torus S1 ×S1 = R2/Z2 by a linear transformation of R2 preserving the lattice of integral points Z2. It turns out that any orientation preserving diﬀeomorphism of the torus is isotopic to a linear mapping so that SL(2, Z) is isomorphic to the mapping class group of the torus. Given a matrix A, one can construct the mapping torus

3 2 2 TA = R /Z × R/{(x, t) ∼ (A(x), t + 1)}.

3 2 2 By definition, TA has R /Z × R as an infinite cyclic cover. The natural 3 1 projection of TA on S = R/Z is a locally trivial fibration, whose fibres are tori. These examples were introduced by Poincaré[109] in his foundational paper on Topology. His motivation was to study examples of manifolds where the homology (that he defined himself) was not sufficient to distinguish non homeomorphic manifolds. Indeed, he defined the fundamental group based 3 3 on these examples. He showed that the fundamental groups of TA and TB are isomorphic if and only if A and B±1 are conjugate in GL(2, Z). In order to connect these manifolds with the geometry of the modular orbifold X = H/PSL(2, Z), one should recall the following construction. Consider the linear action of SL(2, Z) on C2. The open set

•2 Lmarked = {(ω1, ω2) ∈ C | ω1/ω2 ∈ H} is the set of lattices in C marked with a positive basis. The quotient L = Lmarked/SL(2, Z) is the space of (unmarked) lattices in Λ ⊂ C. For such a lattice Λ one deﬁnes

X −4 X −6 g2(Λ) = 60 ω and g3(Λ) = 140 ω . ω∈Λ\{0} ω∈Λ\{0}

It turns out that (g2, g3) maps L biholomorphically onto the complement of 3 2 2 the discriminant locus {(g2, g3)|g2 − 27g3 = 0} ⊂ C . By homogeneity, L is homeomorphic to the product by R+ of the complement of a trefoil knot in 2 2 the 3-sphere {(g2, g3) ∈ C| |g2| + |g3| = 1}. Its fundamental group is the braid group B3, a central extension of SL(2, Z). Note that the maps

(ω1, ω2) ∈ Lmarked → ω1/ω2 ∈ H

120 is SL(2, Z)-equivariant and produces a holomorphic map L → X (with fibres isomorphic to C•). Each lattice Λ ⊂ C defines an elliptic curve C/Z, which is topologically a torus T2. Therefore there is a tautological torus bundle E over L. The 2 action of the fundamental group of the basis, i.e. B3, on the homology Z of the torus fibres, factors through the projection B3 → SL(2, Z) and reduces to canonical action of SL(2, Z) on Z2. Given a loop in L, one can consider its pre-image in E. One gets a 3- 3 manifold which is of course diffeomorphic to some TA. This was the main motivation of Poincaréin his attempt to generalize Riemann’s ideas from complex curves to complex surfaces. If A is an element of SL(2, Z), one can choose a path in SL(2, R) connecting A to I2. This provides a trivialization t of the tangent bundle of 3 TA. We can therefore consider the associated Hirzebruch signature defect : 3 3τ(W )−p1(W, ∂W ) for any 4-manifold W with boundary TA where the class p1 is evaluated relative to its trivialization t on the boundary. We have seen earlier that this does not depend on the choice of W . It is easy to show that the defect does not depend either on the choice of the path SL(2, R) since the effect of a change of path factors through π1(SO) = Z/2Z. Therefore, there is a well defined signature defect

δ : SL(2, Z) → Z. There are several ways to compute this number as a function of A. The first 3 is to find explicitly a manifold WA having TA as its boundary. If A, B are two elements of SL(2, Z), one can construct a 4-manifold WA,B = T (A, B), which is a fibration over a pair of pants (i.e. a disc minus two discs), with fibres diffeomorphic to R2/Z2 and such that the monodromies of the boundary 3 3 3 components are A, B, AB. Therefore, the boundary of WA,B is TAB −TA −TB. 3 3 If we can construct manifolds with boundaries TA and TB, we can glue them 3 to WA,B and this will provide a manifold with boundary TAB. 1 n When A = , it is easy to describe T 3 : it is a principal circle 0 1 A fibration over the torus R2/Z2 with Euler number n. Filling each fibre by 3 a disc, one constructs immediately a 4 manifold WA with boundary TA. 2 2 Alternatively, WA is the unit disc bundle over R /Z in the 2-dimensional vector bundle of Chern class n : the zero section has self intersection n. Since we know that any A in SL(2, Z) can be written uniquely as a product of upper and lower triangular matrices, this provides an explicit construction

121 3 of a manifold WA with boundary TA. It is now easy to compute the intersection form of WA and to get the numerical value of δ(A). In order to state the result, we use the so called Rademacher function.

R : PSL(2, Z) → Z deﬁned in the following way. Every element of PSL(2, Z) can be written as 0 1 k+1 U SU ...U with 0, k+1 ∈ {−1, 0, 1} and j ∈ {−1, 1} for j 6= 0, k + 1. We set k+1 X R(A) = i. i=0 For simplicity, suppose that A is hyperbolic, i.e. that trace (A) > 2. Assume that in the above decomposition 0 = 1 and k+1 = 0, then one gets δ(A) = 4R(A). See [16] for this computation. Another approach, also from [16], is to observe that we know that the coboundary map

δ(AB) − δ(A) − δ(B) = τ(WA,B)

SL(2, Z) × SL(2, Z) → Z extends canonically as a cocycle

SL(2, R) × SL(2, R) → Z. This is indeed the Meyer cocycle that we already discussed. The boundedness of this cocycle and the fact that every element of SL(2, R) is a product of a bounded number of commutators enables us to identify quickly its primitive and to recover this Rademacher formula. A nice example where such a formalism can be applied consists of Hilbert modular surfaces√ (see [68]). For instance, consider the Hilbert modular group ΓD = PSL(2, Z[ D]) (for D positive integer with no square factor, say such that D is not congruent to 1 modulo 4). It embeds in PSL(2√ , R) × PSL(2, R) as a discrete group through the two embeddings of Z[ D] in R. One can therefore construct the quotient WD of H × H by the action of Γ (as an orbifold). This is a 4-dimensional orbifold having a ﬁnite number of ends.√ Each of its ends is associated to an element of the ideal class group of Z[ D] 3 + and is of the form TA × R . Therefore√ WD provides a cobordism between all 3 the TA associated to ideals in Z[ D].

122 6.3 Lens spaces The lens space L(c, a) is the closed, parallelizable 3-dimensional manifold deﬁned for coprime c, a ∈ Z with c > a > 0 by

3 L(c, a) = S /(Z/cZ) where the unit sphere S3 is identiﬁed with {(u, v) ∈ C2 | |u|2 + |v|2 = 1} and Z/cZ acts by (u, v) 7→ (ζu, ζav) with ζ = exp 2πi/c. Lens spaces were introduced by Tietze [136] in 1908. They provided the ﬁrst examples of homotopy equivalent manifolds which are not homeomorphic. The lens space L(c, a) has π1(L(c, a)) = Z/cZ, H∗(L(c, a); Q) = 0 and a genus 1 Heegaard decomposition

1 2 1 2 L(c, a) = S × D ∪A S × D a b for any A = ∈ SL(2, ), corresponding to the symplectic formation c d Z (H−(Z); Z ⊕ {0},L) over Z with

L = A(Z ⊕ {0}) = {(ax, cx) | x ∈ Z} ⊂ Z ⊕ Z.

Conversely, lens spaces and S1 × S2 are the only 3-dimensional manifolds having a Heegaard decomposition of genus 1. As in Hirzebruch, Neumann and Koh [66, p.51] assume that c is odd, a is even, so that there is a unique expression of c/a as an improper continued fraction 1 c/a = χ − 1 1 χ2 − .. χ3 − . 1 − χn = [χ1, χ2, . . . , χn] ∈ Q for χk 6= 0 ∈ 2Z (1 6 k 6 n), corresponding to the iterations of the Euclidean algorithm for gcd(a, c) = 1, as in section 6.1 but with the roles of a, c reversed. The expression is realized by the graph 4-dimensional manifold [ (W (χ), ∂W (χ)) = ( D2 × D2,L(c, a)) n+1

123 obtained by n successive geometric plumbings using the tree An weighted by 1 χk ∈ π1(SO(2)) = π1(S ) = Z

χ1 χ2 χ3 χn−1 χn An : • • • ... • • The manifold W (χ) resolves the singularity at (0, 0, 0) of the 2-dimensional complex space 3 c c−a {(w, z1, z2) ∈ C | w = z1(z2) }. See Jung [72], Hirzebruch [63] and de la Harpe [40]. The intersection matrix of W (χ) is the integral symmetric tridiagonal n × n matrix   χ1 1 0 ... 0 0  1 χ2 1 ... 0 0     0 1 χ ... 0 0   3  φW (χ) = Tri(χ) =  ......  ,  ......     0 0 0 . . . χn−1 1  0 0 0 ... 1 χn which is obtained by the corresponding n successive algebraic plumbings (2.28). The symmetric form over Z n (H2(W (χ); Z), φW (χ)) = (Z , Tri(χ)) is nondegenerate, meaning that det(Tri(χ)) 6= 0. As in Proposition 1.18 the principal minors µk = µk(W (χ)) 6= 0 ∈ Z (1 6 k 6 n) are such that

µk/µk−1 = [χk, χk−1, . . . , χ1] , µk−2 + µk = χkµk−1 (µ−1 = 0, µ0 = 1) Pn and by Theorem 1.19 the signature of W (χ) is τ(W (χ)) = k=1 sign(µk/µk−1). The hypothesis |χk| > 2 gives a simpliﬁcation: Proposition 6.1. [Hirzebruch, Neumann and Koh [66, Lemma 8.12] A n vector χ = (χ1, χ2, . . . , χn) ∈ Z with |χk| > 2 (k = 1, 2, . . . , n) is regular (so that the continued fractions [χk, χk−1, . . . , χ1] ∈ Q are well-deﬁned) with

sign([χk, χk−1, . . . , χ1]) = sign(χk) and n n X X τ(Tri(χ)) = sign([χk, χk−1, . . . , χ1]) = sign(χk) ∈ Z. k=1 k=1

124 Proof. Assume inductively that for some k > 1

(a) |[χk, χk−1, . . . , χ1]| > 1,

(b) sign([χk, χk−1, . . . , χ1]) = sign(χk). By the triangle inequality

−1 |[χk+1, χk, . . . , χ1]| = |χk+1 − [χk, χk−1, . . . , χ1] | −1 > |χk+1| − |[χk, χk−1, . . . , χ1] | > 1 and

−1 sign(χk+1) = sign([χk+1, χk, . . . , χ1] ) = sign([χk+1, χk, . . . , χ1]) giving the inductive step. It now follows from Sylvester’s Theorem 1.19 that

n P τ(Tri(χ)) = sign([χk, χk−1, . . . , χ1]) k=1 n P = sign(χk) ∈ Z. k=1

See Hirzebruch and Mayer [65] and Hirzebruch, Neumann and Koh [66] for early accounts of plumbing.

6.4 Dedekind sums The Dedekind η function deﬁned on H by

∞ Y η(z) = exp(πiz/12) (1 − exp(2πiz)n. n=1 It clearly does not vanish. Its 24th power is the modular ∆ that we mentioned earlier az + b η24( ) = η24(z)(cz + d)12 cz + d which does not vanish. Hence, there is a holomorphic determination of its logarithm ln η on H and taking the logarithm on both sides of the previous identity, we get: az + b 24 ln η( ) = 24 ln η(z) + 6 ln(−(cz + d)2) + 2πiR(A). cz + d

125 for some function R on PSL(2, Z) (the second log in the right hand side is chosen with imaginary part in ] − π, π]). The numerical determination of R(A) has been a challenge, and turned out to be related to many different topics, in particular number theory, topology, and combinatorics. The inspiring paper by M. Atiyah [11] contains an “omnibus theorem” proving that seven definitions of R are equivalent! As the notation suggests, one of the incarnation of R is the previously defined Rademacher function. We now describe a more combinatorial description. For a real number x, one defines ((x)) by ( {x} − 1/2 if x ∈ \ ((x)) = R Z 0 if x ∈ Z with {x} ∈ [0, 1) the fractional part of x ∈ R. Nonadditive:  −1/2 if 0 < {x} + {y} < 1  ((x)) + ((y)) − ((x + y)) = 1/2 if 1 < {x} + {y} < 2  0 if x ∈ Z or y ∈ Z or x + y ∈ Z.

0:5 2 1 0 1 2 3 4 x 0:5

The sawtooth function x 7→ ((x)) was used by Dedekind[41] to count the ±2π jumps in the complex logarithm

iθ log(re ) = log(r) + i(θ + 2nπ) ∈ C(n ∈ Z).

The Dedekind sum for a, c ∈ Z with c 6= 0 is

|c|−1 |c|−1 X k ka 1 X kπ kaπ s(a, c) = = cot cot ∈ c c 4|c| c c Q k=1 k=1 (Rademacher and Grosswald [112], Hirzebruch [64], [68], Hirzebruch and Za- gier [67], Kirby and Melvin [80]).

126

1 a b For ∈ SL(2, ) and χ = (χ , χ , . . . , χ ) ∈ n, W (χ) as in Propo- c d Z 1 2 n Z sition 6.1, the signature defect of W (χ) is given by

n ( X b/3d if c = 0 τ(Tri(χ)) − ( χj)/3 = j=1 (a + d)/3 − 4sign(c)s(a, c) if c 6= 0.

See Barge and Ghys [16] and Kirby and Melvin [80] for proofs and a hyperbolic geometry interpretation, related to the action of SL(2, Z) on the upper half plane H.

127 7 Appendix: Algebraic L-theory of rings with involution and the localization exact sequence

In these notes we have largely worked with forms over commutative rings such as fields and their subrings. However, the applications to the topology of manifolds as well as purely algebraic considerations require the study of forms over an arbitrary ring with involution R such as a group ring Z[π] with π the fundamental group. This has led to the development of the al- n gebraic cobordism groups L (R) (resp. Ln(R)) of n-dimensional f.g. free R-module chain complexes C with a symmetric (resp. quadratic) Poincaré n−∗ ∼ 0 duality φ : H (C) = H∗(C) for n > 0, with L (R) = W (R). The quadratic L-groups L∗(R) are the surgery obstruction groups of Wall [140], which are of central significance in the classification of the homotopy types of manifolds of dimension > 4. The symmetric L-groups L∗(R) were introduced by Mishchenko [100]. We shall be mainly concerned with L∗(R) here. We note some basic properties of the algebraic L-groups:

• a compact oriented n-dimensional manifold M with universal cover Mf n has a symmetric signature σ(M) = (C(Mf), φM ) ∈ L (Z[π1(M)]) which is both a cobordism and a homotopy invariant, and which for n = 4k has image [σ(M)] = τ(M) ∈ L4k(Z) = Z the ordinary signature, ∼ • the quadratic L-groups are 4-periodic, L∗(R) = L∗+4(R), with L2k(R) k (resp. L2k+1(R)) the Witt group of nonsingular (−) -quadratic forms (resp. formations) over R,

• the symmetric L-groups of a Dedekind ring R are 4-periodic, L∗(R) = L∗+4(R), with L2k(R) (resp. L2k+1(R)) the Witt group of nonsingular (−)k-symmetric forms (resp. formations) over R,

∼ ∗ • if 1/2 ∈ R then L∗(R) = L (R), • the symmetric and quadratic L-groups are the same modulo 2-primary ∼ ∗ torsion, L∗(R)[1/2] = L (R)[1/2], so they have the same signature invariants,

• for any morphism f : R → R0 of rings with involution there is deﬁned a long exact sequence of symmetric L-groups

· · · → Ln(R) → Ln(R0) → Ln(f) → Ln−1(R) → ...

128 and similarly for quadratic L-groups,

• for an injective localization f : R → R0 = S−1R of rings with involution inverting a multiplicative subset S−1R of central non-zero divisors with S¯ = S there is deﬁned a long exact sequence of symmetric L-groups

· · · → Ln(R) → Ln(S−1R) →∂ Ln(R,S) → Ln−1(R) → ...

with Ln(R,S) the cobordism group of (n − 1)-dimensional symmet- −1 ric Poincar´ecomplexes (C, φ) over R such that H∗(S C) = 0, and similarly for quadratic L-groups,

We refer to Ranicki [115, 116, 117, 118, 120] for the development of this theory. We shall not attempt to summarize the full theory here! Our aim in this appendix is to extend the theory of forms deﬁned in Section 1 over ﬁelds to forms over a ring with involution R, and to indicate some of the connections between the material in sections 1-6 and the general theory.

7.1 Forms over a ring with involution An involution on a ring R is a function r ∈ R 7→ r¯ ∈ R such that r + s =r ¯+¯s, rs =s. ¯ r¯, 1¯ = 1 and r¯ = r. The three main examples are:

• R is a commutative ring andr ¯ = r is the identity involution,

• R = C with x + iy = x − iy complex conjugation,

• R = Z[π] is a group ring, andg ¯ = g−1 for g ∈ π. Let then R be a ring with involution. The dual of a (left) R-module V is ∗ the (left) R-module V = HomR(V,R) with R acting by

(r, f) ∈ R × V ∗ 7→ (v ∈ V 7→ f(v)r) ∈ V ∗.

The natural R-module morphism

v ∈ V 7→ (f 7→ f(v)) ∈ V ∗∗ is an isomorphism for f.g. free R-modules V , in which case we use it to identify V ∗∗ = V .

129 Let = 1 or −1. An -symmetric form (V, φ) over R is a f.g. free R-module V with a bilinear pairing

φ :(v, w) ∈ V × V 7→ φ(v, w) ∈ R such that φ(w, v) = φ(v, w) ∈ R for all v, w ∈ V . This is equivalent to an R-module morphism

φ : v ∈ V 7→ (w 7→ φ(v, w)) ∈ V ∗ such that φ∗ = φ. For = 1 the form is called symmetric, while for = −1 it is called skew-symmetric.A morphism f :(V, φ) → (V 0, φ0) of -symmetric forms over R is an R-module morphism f : V → V 0 such that f ∗φ0f = φ, or equivalently such that φ0(f(v), f(w)) = φ(v, w) ∈ R. An isomorphism is a morphism such that f : V → V 0 is an R-module isomorphism. ∗ The transpose of an n×n matrix A = (aij) is the n×n matrix A = (¯aji). From the matrix point of view, an -symmetric form (V, φ) with a choice of basis (b1, b2, . . . , bn) for V corresponds to the n × n matrix S = (Sij) such that S∗ = S, with ¯ φ(bi, bj) = Sij = Sji ∈ R. An isomorphism of forms corresponds to linear congruence of matrices: if f :(V, φ) → (V 0, φ0) is an isomorphism of -symmetric forms and f : V → V 0 has invertible matrix A with respect to the bases chosen for V and V 0, then A∗S0A = S. An -symmetric form (V, φ) is nonsingular if the R-module morphism φ : V → V ∗ is an isomorphism. Given an -symmetric form (V, φ) over R and a submodule V 0 ⊆ V deﬁne the orthogonal submodule

V 0⊥ = {v ∈ V | φ(v, v0) = 0 ∈ R for all v0 ∈ V } ⊆ V.

Submodules V1,V2 ⊆ V are orthogonal if

φ(v1, v2) = 0 ∈ R for all v1 ∈ V1, v2 ∈ V2,

⊥ or equivalently V1 ⊆ V2 . A subform (V 0, φ0) ⊆ (V, φ) of an -symmetric form (V, φ) over R is the 0 -symmetric form deﬁned by the restriction φ = φ|V 0 on a direct summand

130 V 0 ⊆ V such that V 0⊥ ⊆ V is a direct summand. Note that V 0⊥ ⊆ V is a direct summand if the R-module morphism φ| : v ∈ V 7→ (v0 7→ φ(v, v0)) ∈ V 0∗ is onto, as is automatically the case if (V, φ) is nonsingular or if R is a field. The radical of (V, φ) is V ⊥ ⊆ V . If im(φ) ⊆ V ∗ is a direct summand (e.g. if R is a field) then V ⊥ ⊆ V is a direct summand, (V/V ⊥, [φ]) is a nonsingular -symmetric form over R, and there are defined isomorphisms

(V, φ) ∼= (V ⊥, 0) ⊕ (im(φ), Φ), (V/V ⊥, [φ]) ∼= (im(φ), Φ) where Φ : im(φ) × im(φ) → R;(φ(v), φ(w)) 7→ φ(v)(w) = φ(w)(v)..

For a symmetric form (V, φ) over R ⊥ τ(V, φ) = τ(V/V , [φ]) ∈ Z Let (V, φ) be an -symmetric form over R.A sublagrangian of an - symmetric form (V, φ) over R is a subform of the type (L, 0) ⊆ (V, φ). Then L ⊆ V is isotropic (self-orthogonal), with L, L⊥ ⊆ V direct summands such that L ⊆ L⊥, with an induced -symmetric form (L⊥/L, [φ]). A lagrangian is a sublagrangian L such that L = L⊥, in which case (V, φ) is nonsingular and dimR(L) = dimR(V )/2. Example 7.1. For any nonsingular -symmetric form (V, φ) over R the diagonal ∆ = {(v, v) ∈ V ⊕ V | v ∈ V } ⊂ V ⊕ V is a lagrangian of (V, φ) ⊕ (V, −φ). Deﬁnition 7.2. 1. The metabolic -symmetric form is the nonsingular -symmetric form over R deﬁned for any -symmetric form (L∗, θ) over R by 0 1 H (L∗, θ) = (L ⊕ L∗, ) θ with 0 1 : L ⊕ L∗ → (L ⊕ L∗)∗; θ (v, f) 7→ ((w, g) 7→ g(v) + f(w) + θ(f)(g))

131 ∗ H(L , θ) has lagrangian L. 2. The hyperbolic -symmetric form is deﬁned for any f.g. free R-module L by 0 1 H (L) = H (L∗, 0) = (L ⊕ L∗, ). 0 Proposition 7.3. 1. The inclusion of a sublagrangian (L, 0) ⊆ (V, φ) in an -symmetric form extends to an isomorphism of forms

∗ ⊥ =∼ H(L , θ) ⊕ (L /L, [φ]) / (V, φ).

In particular, a nonsingular -symmetric form (V, φ) admits a lagrangian L ∗ ∗ if and only if (V, φ) is isomorphic to H(L , θ) for some (L , θ). 2. If 1/2 ∈ R there is defined an isomorphism 1 θ/2 =∼ : H (L∗, θ) / H (L) 0 1 so a nonsingular -symmetric form admits a lagrangian L if and only if it is isomorphic to H(L) for some L. Proof. See Ranicki[116, Prop. 2.2]. We finally get to the definition of the Witt groups of a ring with involution.

Deﬁnition 7.4. The -symmetric Witt group W(R) of a ring with involution R is the group of equivalence classes of nonsingular -symmetric forms (V, φ) over R with (V, φ) ∼ (V 0, φ0) if and only if there exists an isomorphism ∼ ∗ = 0 0 0∗ 0 f :(V, φ) ⊕ H(L , θ) / (V , φ ) ⊕ H(L , θ ) for some -symmetric forms (L∗, θ), ((L0)∗, θ0) over R.

This equivalence relation is called stable isomorphism.

The Witt group W(R) is the quotient of the Grothendieck group of isomorphism classes [V, φ] of nonsingular -symmetric forms over R with the relations [V, φ] = 0 if (V, φ) admits a lagrangian L

132 or equivalently

[V, φ] = [L⊥/L, [φ]] if (V, φ) admits a sublagrangian L.

Addition and inverses are by

0 0 0 0 [V, φ] + [V , φ ] = [V ⊕ V , φ ⊕ φ ], − [V, φ] = [V, −φ] ∈ W(R).

We shall write the symmetric and skew-symmetric Witt groups as

W+1(R) = W (R),W−1(R) = W−(R) and similarly for the hyperbolic and metabolic forms

∗ ∗ H+1(L , θ) = H(L , θ),H+1(L) = H(L), ∗ ∗ H−1(L , θ) = H−(L , θ),H−1(L) = H−(L).

Remark 7.5. By Witt’s Theorem 1.26 for R a field of characteristic 6= 2 (with the identity involution) nonsingular symmetric forms are isomorphic if and only if they are stably isomorphic. This is false for a field of characteristic 2 and also for rings which are not fields: for example if R = F2 or Z the nonsingular symmetric forms H(R), H(R, 1) are stably isomorphic but not isomorphic.

7.2 The localization exact sequence Let R be a ring with involution, and let S ⊂ R be a multiplicative subset of central non-zero divisors such that 1 ∈ S and S = S. The localization of R inverting S is the ring with involution

S−1R = S × R/{(s, r) ∼ (s0, r0) | rs0 = r0s ∈ R} with the equivalence class of (r, s) denoted r/s. The natural morphism of rings with involution r ∈ R 7→ r/1 ∈ S−1R is injective, and the localization exact sequence of Ranicki [117, Chapter III]

· · · → L1(R,S) → L0(R) → L0(S−1R) →∂ L0(R,S) → L−1(R) → ... is deﬁned. The L-group L0(S−1R) = W (S−1R) the Witt group of symmetric −1 forms (V, φ) over R which are S-nonsingular (i.e. S R ⊗R (V, φ) is nonsingular). The L-group L0(R,S) = W (R,S) is the Witt group of nonsingular

133 symmetric linking forms over (R,S), pairs (T, θ) with T a f.g. R-module of homological dimension 1 and θ : T × T → S−1R/R a nonsingular symmetric pairing. The boundary map ∂ : W (S−1R) → W (R,S) is deﬁned by

∂S−1(V, φ) = (coker(φ : V → V ∗), θ :(x, y) 7→ x(φ−1(y))) noting that φ−1 is only deﬁned over S−1R. The function ∂ codiﬁes the expression of the linking form on the odd-dimensional boundary of an even- dimensional manifold in terms of the intersection matrix. Note that for any s, t ∈ S the nonsingular symmetric form (S−1R, s/t) over S−1R has image the linking form ∂(S−1R, s/t) = (R/stR, s/t) over (R,S), with

s/t :(x, y) ∈ R/stR × R/stR 7→ sxy/t¯ ∈ S−1R/R.

If R is a Dedekind ring with quotient ﬁeld (R\{0})−1R = K then for any involution on R and multiplicative subset S ⊂ R as above, the localization of R inverting S is a subring S−1R ⊆ K the Witt group localization exact sequence is

L1(R,S) = 0 → W (R) → W (S−1R) →∂ W (R,S) → L−1(R) → L−1(S−1R) = 0. ∼ L Primary decomposition and d´evissagegive an isomorphism W (R,S) = π W (R/π), with π running over all the involution-invariant prime ideals π / R, and R/π the residue class ﬁelds. Moreover, the function

∞ [ n n n −1 A ∈ Sp(R) = Sp(2n, R) 7→ (H−(R ); R ,A(R )) ∈ L (R) n=1 is a surjective group morphism. Here are some examples of how the Witt group localization exact sequence has appeared in this paper:

• Let R be an integral domain (with the identity involution) and let K = S−1R be the quotient ﬁeld, with S = R\{0}. Given sequences n+1 n (p0, p1, . . . , pn) ∈ S ,(q1, q2, . . . , qn) ∈ S satisfying the Euclidean algorithm recurrences

pk−1 + pk+1 = pkqk (1 6 k 6 n), pn+1 = 0

134 suppose that pn ∈ R is a unit (i.e. p0, p1 are coprime). Deﬁne the tridiagonal S-nonsingular tridiagonal symmetric form (Rn, Tri(q)) over R with   q1 1 0 ... 0  1 q 1 ... 0   2   0 1 q ... 0  Tri(q) =  3  .  ......   . . . . .  0 0 0 . . . qn As in section 1.8 it can be seen that the nonsingular symmetric form n (K , Tri(q)) is diagonalized by p0/p1, p1/p2, . . . , pn−1/pn ∈ K, so that

n n (K , Tri(q)) = ⊕k=1(K, pk−1/pk) ∈ W (K) with

n n X ∂(K , Tri(q)) = (R/(pk−1pk), pk−1/pk) = (R/(p0), p0/p1) ∈ W (R,S). k=1

a b In particular, for any ∈ SL(2, ) we have that a, c ∈ are c d Z Z coprime. There is deﬁned a lens space L(c, a) as in section 6.3, and Tri(q) is the symmetric intersection matrix over Z of a 4-dimensional manifold W with det(Tri(q)) = c and ∂W = L(c, a) constructed by n plumbings.

• The computation of W (Q) in section 1.8 fits into the localization exact sequence ∂ 0 → W (Z) → W (Q) → W (Z,S) → 0 L L with S = Z\{0} ⊂ Z and W (Z,S) = Z2 ⊕ Z4 the Witt group ∞ ∞ of nonsingular linking pairing (T, θ : T × T → Q/Z) on finite abelian groups. In particular, this is the exact sequence used by Milnor and Husemoller [97] to recover the computation that the signature map τ : W (Z) → Z is an isomorphism (Serre [124]). • The computation of W (K(X)) in section 1.8 for any field K for a field of characteristic n 6= 2 fits into the localization exact sequence

0 → W (K[X]) → W (K(X)) →∂ W (K[X],K[X]\{0}) → 0.

135 By a theorem of Karoubi (Ojanguren [105]) W (K[X]) = W (K). By Ranicki [118, Prop. 39.4] W (K[X],K[X]\{0}) is the Witt group of nonsingular symmetric forms (V, φ) over K with a K-linear map X : V → V such that φ(Xv, w) = φ(v, Xw). For any regular degree n polynomial P (X) ∈ K[X] the Witt class (K(X),P (X)) ∈ W (K(X)) has image

∂(K(X),P (X)) = (Kn, S, C(P )) ∈ W (K[X],K[X]\{0})

with S the Hermite matrix of Theorem 1.25 and C(P ) the companion matrix. See [118, Example 39.9] for the computation W (R(X)) = Z ⊕ Z[R] ⊕ Z/2Z[H] in section 1.8 from this point of view. • Given a ring with involution R let the Laurent polynomial extension ring R[z, z−1] have the involutionz ¯ = z−1. For R = Z let P ⊂ Z[z, z−1] be the multiplicative subset of the polynomials p(z) ∈ Z[z, z−1] such that p(z) = p(z−1) ∈ Z[z, z−1]. The complement (X, ∂X) of an n-knot k : Sn ⊂ Sn+2 is equipped with a degree 1 map

1 n+3 n f :(X, ∂X) → S × (D , S )

which is a Z-homology equivalence. The chain complex kernel of f is a Z-acyclic (n + 2)-dimensional symmetric Poincar´ecomplex (C, φ) over −1 Z[z, z ]. The morphism deﬁned on the cobordism group Cn of n-knots

n+3 −1 k ∈ Cn 7→ (C, φ) ∈ L (Z[z, z ],P )

is a surjection for n = 1 and an isomorphism for n > 2. The discussion of the knot signature jumps in section 3.3 can be expressed in terms of the computation of W (R(z)) of the rational function ﬁeld R(z) with involution z = z−1 (cf. Ranicki [118, Chapter 39]), paralleling the formulation in section 1.8 of the theorems of Sturm and Sylvester in terms of the computation of W (R(X)) (X = X).

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